%------------------------------------------------------------------------------
% File     : ITP235_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 01015_065053
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0067_VEBT_InsertCorrectness_01015_065053 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11974 (3223 unt;1571 typ;   0 def)
%            Number of atoms       : 26914 (8168 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 18116 (2819   ~; 554   |;1594   &)
%                                         (3061 <=>;10088  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       : 1995 ( 958 fml;1037 var)
%            Number of X terms     :  625 (   0  []; 562 ite;  63 let)
%            Number of types       :  262 ( 261 usr)
%            Number of type conns  : 1453 ( 903   >; 550   *;   0   +;   0  <<)
%            Number of predicates  :  132 ( 129 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1192 (1192 usr; 418 con; 0-8 aty)
%            Number of variables   : 23881 (23028   !; 853   ?;23881   :)
% 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
%            2022-02-17 21:53:14.717
%------------------------------------------------------------------------------
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    fun_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Rat__Orat_J,type,
    fun_o_rat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Num__Onum_J,type,
    fun_o_num: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
    fun_o_nat: $tType ).

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

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

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

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

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

tff(ty_n_t__String__Oliteral,type,
    literal: $tType ).

tff(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

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

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

tff(ty_n_t__Real__Oreal,type,
    real: $tType ).

tff(ty_n_t__Rat__Orat,type,
    rat: $tType ).

tff(ty_n_t__Num__Onum,type,
    num: $tType ).

tff(ty_n_t__Nat__Onat,type,
    nat: $tType ).

tff(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (1310)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_Extended_enat > fun_Extended_enat_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_Extended_enat_o > fun_fu9148622857633396327enat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_Extended_enat_o > fun_fu9148622857633396327enat_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_Extended_enat > fun_se8123728892507317397enat_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_Ex239764483819389678enat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: int > fun_int_fun_int_o ).

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

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_Pr4811707699266497531nteger > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_int_int ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_real_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_real_o ).

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

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: int > fun_int_fun_int_o ).

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

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_real_real ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_re4238438383274954274t_real > fun_real_real ).

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

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_nat_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: real > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: nat > fun_li6914387844523535256_nat_o ).

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

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_real_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_nat_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_fun_nat_rat_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_na1476630412504255114at_nat ).

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

tff(sy_c_ATP_058Lamm__gg____,type,
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tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: nat > fun_nat_set_nat ).

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

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
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tff(sy_c_ATP_058Lamm__gn____,type,
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tff(sy_c_ATP_058Lamm__go____,type,
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tff(sy_c_ATP_058Lamm__gp____,type,
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tff(sy_c_ATP_058Lamm__gq____,type,
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tff(sy_c_ATP_058Lamm__gr____,type,
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tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__gt____,type,
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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,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
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tff(sy_c_Binomial_Obinomial,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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tff(sy_c_Bit__Operations_Oconcat__bit,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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tff(sy_c_Code__Numeral_Obit__cut__integer,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Complex_Ocis,type,
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tff(sy_c_Complex_Ocomplex_ORe,type,
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tff(sy_c_Divides_Oeucl__rel__int,type,
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tff(sy_c_Extended__Nat_OeSuc,type,
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tff(sy_c_Extended__Nat_Oenat,type,
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    lattic4004264746738138117at_num: ( fun_nat_num * set_nat ) > nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
    lattic6811802900495863747at_rat: ( fun_nat_rat * set_nat ) > nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
    lattic488527866317076247t_real: ( fun_nat_real * set_nat ) > nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
    lattic4420706379359479199al_rat: ( fun_real_rat * set_real ) > real ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
    lattic8440615504127631091l_real: ( fun_real_real * set_real ) > real ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_Eo,type,
    lattic4107685809792843317_fin_o: fun_set_o_o ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    lattic9128708996678779395er_nat: set_filter_nat > filter_nat ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Int__Oint,type,
    lattic5235898064620869839in_int: set_int > int ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
    lattic5238388535129920115in_nat: set_nat > nat ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Rat__Orat,type,
    lattic4603258475043424379in_rat: set_rat > rat ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Real__Oreal,type,
    lattic2677971596711400399n_real: set_real > real ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_I_Eo_J,type,
    lattic3298725636695511317_set_o: set_set_o > set_o ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Int__Oint_J,type,
    lattic8060154151401097861et_int: set_set_int > set_int ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3014633134055518761et_nat: set_set_nat > set_nat ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001_Eo,type,
    lattic1508158080041050831_fin_o: fun_set_o_o ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    lattic5930898082463196905er_nat: set_filter_nat > filter_nat ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Int__Oint,type,
    lattic1091506334969745077in_int: set_int > int ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
    lattic1093996805478795353in_nat: set_nat > nat ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Rat__Orat,type,
    lattic458866745392299617in_rat: set_rat > rat ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Real__Oreal,type,
    lattic8928443293348198069n_real: set_real > real ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_I_Eo_J,type,
    lattic3158155371183623599_set_o: set_set_o > set_o ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Int__Oint_J,type,
    lattic8880645941091133547et_int: set_set_int > set_int ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3835124923745554447et_nat: set_set_nat > set_nat ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lattic1541023418247406232at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( fun_nat_real * filter_nat ) > $o ).

tff(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

tff(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: ( list_int * list_int ) > list_int ).

tff(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: ( list_nat * list_nat ) > list_nat ).

tff(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

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

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

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( fun_nat_nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

tff(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
    set_Extended_enat2: list_Extended_enat > set_Extended_enat ).

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

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

tff(sy_c_List_Olist_Oset_001t__Product____Type__Ounit,type,
    set_Product_unit2: list_Product_unit > set_Product_unit ).

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

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

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

tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > nat ).

tff(sy_c_List_Olist__update_001_Eo,type,
    list_update_o: ( list_o * nat * $o ) > list_o ).

tff(sy_c_List_Olist__update_001t__Extended____Nat__Oenat,type,
    list_u3071683517702156500d_enat: ( list_Extended_enat * nat * extended_enat ) > list_Extended_enat ).

tff(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: ( list_int * nat * int ) > list_int ).

tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: ( list_nat * nat * nat ) > list_nat ).

tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
    list_update_set_nat: ( list_set_nat * nat * set_nat ) > list_set_nat ).

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

tff(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > fun_nat_o ).

tff(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: ( list_Code_integer * nat ) > code_integer ).

tff(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
    nth_Extended_enat: ( list_Extended_enat * nat ) > extended_enat ).

tff(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > fun_nat_int ).

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    nth_Pr8522763379788166057eger_o: ( list_P8526636022914148096eger_o * nat ) > produc6271795597528267376eger_o ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    nth_Pr4439495888332055232nt_int: ( list_P5707943133018811711nt_int * nat ) > product_prod_int_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr3474266648193625910T_VEBT: ( list_P7524865323317820941T_VEBT * nat ) > produc1531783533982839933T_VEBT ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    nth_Pr6837108013167703752BT_int: ( list_P4547456442757143711BT_int * nat ) > produc4894624898956917775BT_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > fun_nat_real ).

tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: ( list_set_nat * nat ) > set_nat ).

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

tff(sy_c_List_Onull_001t__Nat__Onat,type,
    null_nat: fun_list_nat_o ).

tff(sy_c_List_Onull_001t__VEBT____Definitions__OVEBT,type,
    null_VEBT_VEBT: fun_list_VEBT_VEBT_o ).

tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: ( list_int * list_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc662631939642741121T_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: ( list_VEBT_VEBT * list_int ) > list_P4547456442757143711BT_int ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_OremoveAll_001_Eo,type,
    removeAll_o: ( $o * list_o ) > list_o ).

tff(sy_c_List_OremoveAll_001t__Extended____Nat__Oenat,type,
    remove8473807646742367858d_enat: ( extended_enat * list_Extended_enat ) > list_Extended_enat ).

tff(sy_c_List_OremoveAll_001t__Int__Oint,type,
    removeAll_int: ( int * list_int ) > list_int ).

tff(sy_c_List_OremoveAll_001t__Nat__Onat,type,
    removeAll_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_OremoveAll_001t__Real__Oreal,type,
    removeAll_real: ( real * list_real ) > list_real ).

tff(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
    removeAll_set_nat: ( set_nat * list_set_nat ) > list_set_nat ).

tff(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
    removeAll_VEBT_VEBT: ( vEBT_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

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

tff(sy_c_List_Orotate1_001_Eo,type,
    rotate1_o: list_o > list_o ).

tff(sy_c_List_Orotate1_001t__Int__Oint,type,
    rotate1_int: list_int > list_int ).

tff(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

tff(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
    rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: ( fun_int_fun_int_o * list_int ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_o > fun_list_nat_o ).

tff(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: ( nat * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Ounion_001_Eo,type,
    union_o: ( list_o * list_o ) > list_o ).

tff(sy_c_List_Ounion_001t__Int__Oint,type,
    union_int: ( list_int * list_int ) > list_int ).

tff(sy_c_List_Ounion_001t__Nat__Onat,type,
    union_nat: ( list_nat * list_nat ) > list_nat ).

tff(sy_c_List_Ounion_001t__VEBT____Definitions__OVEBT,type,
    union_VEBT_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

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

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_fun_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

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

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    size_s3941691890525107288d_enat: list_Extended_enat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

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

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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_I_Eo_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_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____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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    ord_le3650406611588909774_nat_o: ( fun_Pr2028404973800837934_nat_o * fun_Pr2028404973800837934_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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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,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: fun_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: fun_se2621318847353125966_set_o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_ma4205026669011143323d_enat: ( set_Extended_enat * set_Extended_enat ) > set_Extended_enat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: fun_se6573205548028396576et_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: fun_se1909708154429037836et_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_Omax_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_max_set_set_nat: ( set_set_nat * set_set_nat ) > set_set_nat ).

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

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

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

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: fun_real_real > $o ).

tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_nat_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: ( code_integer * 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_001_Eo,type,
    produc6677183202524767010eger_o: ( code_integer * $o ) > produc6271795597528267376eger_o ).

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

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
    product_Pair_int_o: ( int * $o ) > product_prod_int_o ).

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__Int__Oint_001t__Nat__Onat,type,
    product_Pair_int_nat: ( int * nat ) > product_prod_int_nat ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: ( product_prod_nat_nat * product_prod_nat_nat ) > produc859450856879609959at_nat ).

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

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: ( vEBT_VEBT * nat ) > produc9072475918466114483BT_nat ).

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

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

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: ( fun_Co8499238673658658103nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_int_fun_int_o > fun_Pr3238374608701176162_int_o ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: fun_nat_fun_nat_o > fun_Pr4796943543331071514_nat_o ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc6395135316500328296_nat_o: fun_na5914473681949881652_nat_o > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    produc5414030515140494994real_o: fun_real_fun_real_o > fun_Pr4814023175418636514real_o ).

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

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

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

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: product_prod_int_int > rat ).

tff(sy_c_Rat_OFract,type,
    fract: int > fun_int_rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: fun_rat_real ).

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

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr7537129237977221859_rat_o ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_o ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Real_OReal,type,
    real2: fun_nat_rat > real ).

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

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu1771228864347509049real_o ).

tff(sy_c_Real_Opositive,type,
    positive2: fun_real_o ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu6714665071556184448_rat_o ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun_real_fun_nat_rat ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun_nat_rat > $o ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear__axioms_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    real_V7139242839884736329omplex: fun_complex_complex > $o ).

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

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: fun_re7681423839302404990l_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

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

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: fun_real_real ).

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

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: fun_nat_fun_nat_o ).

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

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: fun_re7681423839302404990l_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co2081214437782444048eger_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: fun_co1148497455491383662plex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

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

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Set_OCollect_001_Eo,type,
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tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_OCollect_001t__Int__Oint,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_o > set_list_nat ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001_Eo,type,
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tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Oinsert_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
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tff(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
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tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
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tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Ois__empty_001_Eo,type,
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tff(sy_c_Set_Ois__empty_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Ois__empty_001t__Int__Oint,type,
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tff(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
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tff(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
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tff(sy_c_Set_Ois__empty_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Ois__empty_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Ois__singleton_001_Eo,type,
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tff(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Ois__singleton_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
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tff(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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tff(sy_c_Set_Ois__singleton_001t__Product____Type__Ounit,type,
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tff(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
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tff(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oremove_001_Eo,type,
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tff(sy_c_Set_Oremove_001t__Extended____Nat__Oenat,type,
    remove_Extended_enat: ( extended_enat * set_Extended_enat ) > set_Extended_enat ).

tff(sy_c_Set_Oremove_001t__Int__Oint,type,
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tff(sy_c_Set_Oremove_001t__Nat__Onat,type,
    remove_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oremove_001t__Real__Oreal,type,
    remove_real: ( real * set_real ) > set_real ).

tff(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
    remove_set_nat: ( set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: fun_set_o_o ).

tff(sy_c_Set_Othe__elem_001t__Extended____Nat__Oenat,type,
    the_el319773668273709403d_enat: set_Extended_enat > extended_enat ).

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

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

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

tff(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
    the_elem_set_nat: set_set_nat > set_nat ).

tff(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: ( nat * nat ) > set_nat ).

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or370866239135849197et_int: ( set_int * set_int ) > set_set_int ).

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: nat > set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: ( nat * nat ) > set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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

tff(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: ( literal * fun_Pr1792556611952799308t_real ) > real ).

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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

tff(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).

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

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

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: fun_nat_real > $o ).

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

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__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_Opowr__real,type,
    powr_real2: 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__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

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

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr1263128411319240110enat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

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

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( fun_Pr9018117161326028078_int_o * product_prod_int_int ) > $o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: ( fun_Pr1263128411319240110enat_o * produc7272778201969148633d_enat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr8877776288780692068_nat_o * produc9072475918466114483BT_nat ) > $o ).

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

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

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

tff(sy_c_aa_001_Eo_001t__Code____Numeral__Ointeger,type,
    aa_o_Code_integer: ( fun_o_Code_integer * $o ) > code_integer ).

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

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

tff(sy_c_aa_001_Eo_001t__Num__Onum,type,
    aa_o_num: ( fun_o_num * $o ) > num ).

tff(sy_c_aa_001_Eo_001t__Rat__Orat,type,
    aa_o_rat: ( fun_o_rat * $o ) > rat ).

tff(sy_c_aa_001_Eo_001t__Real__Oreal,type,
    aa_o_real: ( fun_o_real * $o ) > real ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_M_Eo_J,type,
    aa_o_fun_o_o: ( fun_o_fun_o_o * $o ) > fun_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_o_fun_real_o: ( fun_o_fun_real_o * $o ) > fun_real_o ).

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

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

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

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    aa_Cod1096751537565085541eger_o: ( fun_Co3544761417380347684eger_o * code_integer ) > fun_Co1766037875893669450eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001_Eo,type,
    aa_complex_o: ( fun_complex_o * complex ) > $o ).

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Num__Onum,type,
    aa_complex_num: ( fun_complex_num * complex ) > num ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    aa_complex_rat: ( fun_complex_rat * complex ) > rat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    aa_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_com5808801942499186519plex_o: ( fun_co1148497455491383662plex_o * complex ) > fun_complex_o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001_Eo,type,
    aa_Extended_enat_o: ( fun_Extended_enat_o * extended_enat ) > $o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    aa_Extended_enat_rat: ( fun_Ex4152069839811072081at_rat * extended_enat ) > rat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    aa_Ext2541572767299617110t_real: ( fun_Ex1451620480498224165t_real * extended_enat ) > real ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_I_Eo_M_Eo_J,type,
    aa_Ext1181273229397116269un_o_o: ( fun_Ex3734807115953967292un_o_o * extended_enat ) > fun_o_o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
    aa_Ext6368937152127170207enat_o: ( fun_Ex239764483819389678enat_o * extended_enat ) > fun_Extended_enat_o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    aa_Ext9056830302645520696unit_o: ( fun_Ex5682132948637686023unit_o * extended_enat ) > fun_Product_unit_o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_Ext8505340146412899099real_o: ( fun_Ex8943761750289878562real_o * extended_enat ) > fun_real_o ).

tff(sy_c_aa_001t__Filter__Ofilter_It__Nat__Onat_J_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    aa_fil4124504356706564990er_nat: ( fun_fi2406938933853765693er_nat * filter_nat ) > filter_nat ).

tff(sy_c_aa_001t__Filter__Ofilter_It__Nat__Onat_J_001t__fun_It__Filter__Ofilter_It__Nat__Onat_J_Mt__Filter__Ofilter_It__Nat__Onat_J_J,type,
    aa_fil1282661720248192731er_nat: ( fun_fi7557781610047541018er_nat * filter_nat ) > fun_fi2406938933853765693er_nat ).

tff(sy_c_aa_001t__Int__Oint_001_Eo,type,
    aa_int_o: ( fun_int_o * int ) > $o ).

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

tff(sy_c_aa_001t__Int__Oint_001t__Nat__Onat,type,
    aa_int_nat: ( fun_int_nat * int ) > nat ).

tff(sy_c_aa_001t__Int__Oint_001t__Num__Onum,type,
    aa_int_num: ( fun_int_num * int ) > num ).

tff(sy_c_aa_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Rat__Orat,type,
    aa_int_rat: ( fun_int_rat * int ) > rat ).

tff(sy_c_aa_001t__Int__Oint_001t__Real__Oreal,type,
    aa_int_real: ( fun_int_real * int ) > real ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_I_Eo_M_Eo_J,type,
    aa_int_fun_o_o: ( fun_int_fun_o_o * int ) > fun_o_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_int_fun_int_o: ( fun_int_fun_int_o * int ) > fun_int_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_int2464330468366073224_int_o: ( fun_in5570076352459579423_int_o * int ) > fun_int_fun_int_o ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_int_fun_real_o: ( fun_int_fun_real_o * int ) > fun_real_o ).

tff(sy_c_aa_001t__List__Olist_I_Eo_J_001_Eo,type,
    aa_list_o_o: ( fun_list_o_o * list_o ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    aa_list_nat_o: ( fun_list_nat_o * list_nat ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001_Eo,type,
    aa_list_VEBT_VEBT_o: ( fun_list_VEBT_VEBT_o * list_VEBT_VEBT ) > $o ).

tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_J_J_J,type,
    aa_lis144135085560928841_nat_o: ( fun_li6914387844523535256_nat_o * list_VEBT_VEBT ) > fun_VE2573032740582156747_nat_o ).

tff(sy_c_aa_001t__Nat__Onat_001_Eo,type,
    aa_nat_o: ( fun_nat_o * nat ) > $o ).

tff(sy_c_aa_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
    aa_nat_Extended_enat: ( fun_na8797796813480665199d_enat * nat ) > extended_enat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Int__Oint,type,
    aa_nat_int: ( fun_nat_int * nat ) > int ).

tff(sy_c_aa_001t__Nat__Onat_001t__Nat__Onat,type,
    aa_nat_nat: ( fun_nat_nat * nat ) > nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__Num__Onum,type,
    aa_nat_num: ( fun_nat_num * nat ) > num ).

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tff(sy_c_aa_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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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__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

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

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

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    member1379723562493234055eger_o: ( produc6271795597528267376eger_o * set_Pr448751882837621926eger_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: ( produc8923325533196201883nteger * set_Pr4811707699266497531nteger ) > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Ounit,type,
    member_Product_unit: ( product_unit * set_Product_unit ) > $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_I_Eo_J,type,
    member_set_o: ( set_o * set_set_o ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: ( set_int * set_set_int ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_set_set_nat_o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_t,type,
    t: vEBT_VEBT ).

tff(sy_v_x,type,
    x: nat ).

tff(sy_v_y____,type,
    y: nat ).

% Relevant facts (10387)
tff(fact_0__092_060open_062y_A_092_060in_062_Aset__vebt_H_A_Ivebt__insert_At_Ax_J_092_060close_062,axiom,
    aa_set_nat_o(member_nat(y),vEBT_VEBT_set_vebt(vEBT_vebt_insert(t,x))) ).

% \<open>y \<in> set_vebt' (vebt_insert t x)\<close>
tff(fact_1_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_2_Un__insert__left,axiom,
    ! [A: set_nat,B: set_set_nat,C: set_set_nat] : sup_sup_set_set_nat(insert_set_nat(A,B),C) = insert_set_nat(A,sup_sup_set_set_nat(B,C)) ).

% Un_insert_left
tff(fact_3_Un__insert__left,axiom,
    ! [A: real,B: set_real,C: set_real] : sup_sup_set_real(insert_real(A,B),C) = insert_real(A,sup_sup_set_real(B,C)) ).

% Un_insert_left
tff(fact_4_Un__insert__left,axiom,
    ! [A: extended_enat,B: set_Extended_enat,C: set_Extended_enat] : sup_su4489774667511045786d_enat(insert_Extended_enat(A,B),C) = insert_Extended_enat(A,sup_su4489774667511045786d_enat(B,C)) ).

% Un_insert_left
tff(fact_5_Un__insert__left,axiom,
    ! [A: int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A,B)),C) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) ).

% Un_insert_left
tff(fact_6_Un__insert__left,axiom,
    ! [A: $o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,insert_o((A),B)),C) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) ).

% Un_insert_left
tff(fact_7_Un__insert__left,axiom,
    ! [A: nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,B)),C) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).

% Un_insert_left
tff(fact_8_Un__insert__right,axiom,
    ! [A2: set_set_nat,A: set_nat,B: set_set_nat] : sup_sup_set_set_nat(A2,insert_set_nat(A,B)) = insert_set_nat(A,sup_sup_set_set_nat(A2,B)) ).

% Un_insert_right
tff(fact_9_Un__insert__right,axiom,
    ! [A2: set_real,A: real,B: set_real] : sup_sup_set_real(A2,insert_real(A,B)) = insert_real(A,sup_sup_set_real(A2,B)) ).

% Un_insert_right
tff(fact_10_Un__insert__right,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] : sup_su4489774667511045786d_enat(A2,insert_Extended_enat(A,B)) = insert_Extended_enat(A,sup_su4489774667511045786d_enat(A2,B)) ).

% Un_insert_right
tff(fact_11_Un__insert__right,axiom,
    ! [A2: set_int,A: int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),insert_int(A,B)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) ).

% Un_insert_right
tff(fact_12_Un__insert__right,axiom,
    ! [A2: set_o,A: $o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),insert_o((A),B)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) ).

% Un_insert_right
tff(fact_13_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),insert_nat(A,B)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) ).

% Un_insert_right
tff(fact_14_Un__empty,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( sup_sup_set_set_nat(A2,B) = bot_bot_set_set_nat )
    <=> ( ( A2 = bot_bot_set_set_nat )
        & ( B = bot_bot_set_set_nat ) ) ) ).

% Un_empty
tff(fact_15_Un__empty,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( sup_sup_set_real(A2,B) = bot_bot_set_real )
    <=> ( ( A2 = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% Un_empty
tff(fact_16_Un__empty,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( sup_su4489774667511045786d_enat(A2,B) = bot_bo7653980558646680370d_enat )
    <=> ( ( A2 = bot_bo7653980558646680370d_enat )
        & ( B = bot_bo7653980558646680370d_enat ) ) ) ).

% Un_empty
tff(fact_17_Un__empty,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_18_Un__empty,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_19_Un__empty,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = bot_bot_set_o )
    <=> ( ( A2 = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_20_singletonI,axiom,
    ! [A: extended_enat] : aa_set3337841544853333262enat_o(member_Extended_enat(A),insert_Extended_enat(A,bot_bo7653980558646680370d_enat)) ).

% singletonI
tff(fact_21_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real(A),insert_real(A,bot_bot_set_real)) ).

% singletonI
tff(fact_22_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat)) ).

% singletonI
tff(fact_23_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),insert_nat(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_24_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_25_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),insert_o((A),bot_bot_set_o)) ).

% singletonI
tff(fact_26_sup__bot__left,axiom,
    ! [Xa: set_set_nat] : sup_sup_set_set_nat(bot_bot_set_set_nat,Xa) = Xa ).

% sup_bot_left
tff(fact_27_sup__bot__left,axiom,
    ! [Xa: set_real] : sup_sup_set_real(bot_bot_set_real,Xa) = Xa ).

% sup_bot_left
tff(fact_28_sup__bot__left,axiom,
    ! [Xa: set_Extended_enat] : sup_su4489774667511045786d_enat(bot_bo7653980558646680370d_enat,Xa) = Xa ).

% sup_bot_left
tff(fact_29_sup__bot__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),Xa) = Xa ).

% sup_bot_left
tff(fact_30_sup__bot__left,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),Xa) = Xa ).

% sup_bot_left
tff(fact_31_sup__bot__left,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),Xa) = Xa ).

% sup_bot_left
tff(fact_32_sup__bot__left,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,bot_bot_filter_nat),Xa) = Xa ).

% sup_bot_left
tff(fact_33_sup__bot__right,axiom,
    ! [Xa: set_set_nat] : sup_sup_set_set_nat(Xa,bot_bot_set_set_nat) = Xa ).

% sup_bot_right
tff(fact_34_sup__bot__right,axiom,
    ! [Xa: set_real] : sup_sup_set_real(Xa,bot_bot_set_real) = Xa ).

% sup_bot_right
tff(fact_35_sup__bot__right,axiom,
    ! [Xa: set_Extended_enat] : sup_su4489774667511045786d_enat(Xa,bot_bo7653980558646680370d_enat) = Xa ).

% sup_bot_right
tff(fact_36_sup__bot__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),bot_bot_set_nat) = Xa ).

% sup_bot_right
tff(fact_37_sup__bot__right,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),bot_bot_set_int) = Xa ).

% sup_bot_right
tff(fact_38_sup__bot__right,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),bot_bot_set_o) = Xa ).

% sup_bot_right
tff(fact_39_sup__bot__right,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),bot_bot_filter_nat) = Xa ).

% sup_bot_right
tff(fact_40_bot__eq__sup__iff,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] :
      ( ( bot_bot_set_set_nat = sup_sup_set_set_nat(Xa,Ya) )
    <=> ( ( Xa = bot_bot_set_set_nat )
        & ( Ya = bot_bot_set_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_41_bot__eq__sup__iff,axiom,
    ! [Xa: set_real,Ya: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(Xa,Ya) )
    <=> ( ( Xa = bot_bot_set_real )
        & ( Ya = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
tff(fact_42_bot__eq__sup__iff,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = sup_su4489774667511045786d_enat(Xa,Ya) )
    <=> ( ( Xa = bot_bo7653980558646680370d_enat )
        & ( Ya = bot_bo7653980558646680370d_enat ) ) ) ).

% bot_eq_sup_iff
tff(fact_43_bot__eq__sup__iff,axiom,
    ! [Xa: set_nat,Ya: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) )
    <=> ( ( Xa = bot_bot_set_nat )
        & ( Ya = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_44_bot__eq__sup__iff,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) )
    <=> ( ( Xa = bot_bot_set_int )
        & ( Ya = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_45_bot__eq__sup__iff,axiom,
    ! [Xa: set_o,Ya: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) )
    <=> ( ( Xa = bot_bot_set_o )
        & ( Ya = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_46_bot__eq__sup__iff,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] :
      ( ( bot_bot_filter_nat = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) )
    <=> ( ( Xa = bot_bot_filter_nat )
        & ( Ya = bot_bot_filter_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_47_sup__eq__bot__iff,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] :
      ( ( sup_sup_set_set_nat(Xa,Ya) = bot_bot_set_set_nat )
    <=> ( ( Xa = bot_bot_set_set_nat )
        & ( Ya = bot_bot_set_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_48_sup__eq__bot__iff,axiom,
    ! [Xa: set_real,Ya: set_real] :
      ( ( sup_sup_set_real(Xa,Ya) = bot_bot_set_real )
    <=> ( ( Xa = bot_bot_set_real )
        & ( Ya = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
tff(fact_49_sup__eq__bot__iff,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] :
      ( ( sup_su4489774667511045786d_enat(Xa,Ya) = bot_bo7653980558646680370d_enat )
    <=> ( ( Xa = bot_bo7653980558646680370d_enat )
        & ( Ya = bot_bo7653980558646680370d_enat ) ) ) ).

% sup_eq_bot_iff
tff(fact_50_sup__eq__bot__iff,axiom,
    ! [Xa: set_nat,Ya: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = bot_bot_set_nat )
    <=> ( ( Xa = bot_bot_set_nat )
        & ( Ya = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_51_sup__eq__bot__iff,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = bot_bot_set_int )
    <=> ( ( Xa = bot_bot_set_int )
        & ( Ya = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_52_sup__eq__bot__iff,axiom,
    ! [Xa: set_o,Ya: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = bot_bot_set_o )
    <=> ( ( Xa = bot_bot_set_o )
        & ( Ya = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_53_sup__eq__bot__iff,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] :
      ( ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = bot_bot_filter_nat )
    <=> ( ( Xa = bot_bot_filter_nat )
        & ( Ya = bot_bot_filter_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_54_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_55_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_56_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_57_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: filter_nat,B2: filter_nat] :
      ( ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = bot_bot_filter_nat )
    <=> ( ( A = bot_bot_filter_nat )
        & ( B2 = bot_bot_filter_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_58_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ( sup_sup_set_set_nat(A,B2) = bot_bot_set_set_nat )
    <=> ( ( A = bot_bot_set_set_nat )
        & ( B2 = bot_bot_set_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_59_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_real,B2: set_real] :
      ( ( sup_sup_set_real(A,B2) = bot_bot_set_real )
    <=> ( ( A = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_60_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Extended_enat,B2: set_Extended_enat] :
      ( ( sup_su4489774667511045786d_enat(A,B2) = bot_bo7653980558646680370d_enat )
    <=> ( ( A = bot_bo7653980558646680370d_enat )
        & ( B2 = bot_bo7653980558646680370d_enat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_61_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_62_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A) = A ).

% sup_bot.left_neutral
tff(fact_63_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),A) = A ).

% sup_bot.left_neutral
tff(fact_64_sup__bot_Oleft__neutral,axiom,
    ! [A: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,bot_bot_filter_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_65_sup__bot_Oleft__neutral,axiom,
    ! [A: set_set_nat] : sup_sup_set_set_nat(bot_bot_set_set_nat,A) = A ).

% sup_bot.left_neutral
tff(fact_66_sup__bot_Oleft__neutral,axiom,
    ! [A: set_real] : sup_sup_set_real(bot_bot_set_real,A) = A ).

% sup_bot.left_neutral
tff(fact_67_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Extended_enat] : sup_su4489774667511045786d_enat(bot_bo7653980558646680370d_enat,A) = A ).

% sup_bot.left_neutral
tff(fact_68_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_69_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) )
    <=> ( ( A = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_70_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B2: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) )
    <=> ( ( A = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_71_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: filter_nat,B2: filter_nat] :
      ( ( bot_bot_filter_nat = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) )
    <=> ( ( A = bot_bot_filter_nat )
        & ( B2 = bot_bot_filter_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_72_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = sup_sup_set_set_nat(A,B2) )
    <=> ( ( A = bot_bot_set_set_nat )
        & ( B2 = bot_bot_set_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_73_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_real,B2: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(A,B2) )
    <=> ( ( A = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_74_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Extended_enat,B2: set_Extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = sup_su4489774667511045786d_enat(A,B2) )
    <=> ( ( A = bot_bo7653980558646680370d_enat )
        & ( B2 = bot_bo7653980558646680370d_enat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_75_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X: list_nat] : ~ aa_list_nat_o(P,X) ) ).

% empty_Collect_eq
tff(fact_76_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X: nat] : ~ aa_nat_o(P,X) ) ).

% empty_Collect_eq
tff(fact_77_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X: int] : ~ aa_int_o(P,X) ) ).

% empty_Collect_eq
tff(fact_78_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P) )
    <=> ! [X: $o] : ~ aa_o_o(P,(X)) ) ).

% empty_Collect_eq
tff(fact_79_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X: set_nat] : ~ aa_set_nat_o(P,X) ) ).

% empty_Collect_eq
tff(fact_80_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X: real] : ~ aa_real_o(P,X) ) ).

% empty_Collect_eq
tff(fact_81_empty__Collect__eq,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ( bot_bo7653980558646680370d_enat = collec4429806609662206161d_enat(P) )
    <=> ! [X: extended_enat] : ~ aa_Extended_enat_o(P,X) ) ).

% empty_Collect_eq
tff(fact_82_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X: list_nat] : ~ aa_list_nat_o(P,X) ) ).

% Collect_empty_eq
tff(fact_83_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X: nat] : ~ aa_nat_o(P,X) ) ).

% Collect_empty_eq
tff(fact_84_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X: int] : ~ aa_int_o(P,X) ) ).

% Collect_empty_eq
tff(fact_85_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ! [X: $o] : ~ aa_o_o(P,(X)) ) ).

% Collect_empty_eq
tff(fact_86_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X: set_nat] : ~ aa_set_nat_o(P,X) ) ).

% Collect_empty_eq
tff(fact_87_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X: real] : ~ aa_real_o(P,X) ) ).

% Collect_empty_eq
tff(fact_88_Collect__empty__eq,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ( collec4429806609662206161d_enat(P) = bot_bo7653980558646680370d_enat )
    <=> ! [X: extended_enat] : ~ aa_Extended_enat_o(P,X) ) ).

% Collect_empty_eq
tff(fact_89_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X: nat] : ~ aa_set_nat_o(member_nat(X),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_90_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X: int] : ~ aa_set_int_o(member_int(X),A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_91_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ! [X: $o] : ~ aa_set_o_o(member_o((X)),A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_92_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X),A2)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_93_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X: real] : ~ aa_set_real_o(member_real(X),A2)
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_94_all__not__in__conv,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [X: extended_enat] : ~ aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% all_not_in_conv
tff(fact_95_empty__iff,axiom,
    ! [C2: nat] : ~ aa_set_nat_o(member_nat(C2),bot_bot_set_nat) ).

% empty_iff
tff(fact_96_empty__iff,axiom,
    ! [C2: int] : ~ aa_set_int_o(member_int(C2),bot_bot_set_int) ).

% empty_iff
tff(fact_97_empty__iff,axiom,
    ! [C2: $o] : ~ aa_set_o_o(member_o((C2)),bot_bot_set_o) ).

% empty_iff
tff(fact_98_empty__iff,axiom,
    ! [C2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(C2),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_99_empty__iff,axiom,
    ! [C2: real] : ~ aa_set_real_o(member_real(C2),bot_bot_set_real) ).

% empty_iff
tff(fact_100_empty__iff,axiom,
    ! [C2: extended_enat] : ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),bot_bo7653980558646680370d_enat) ).

% empty_iff
tff(fact_101_insert__absorb2,axiom,
    ! [Xa: nat,A2: set_nat] : insert_nat(Xa,insert_nat(Xa,A2)) = insert_nat(Xa,A2) ).

% insert_absorb2
tff(fact_102_insert__absorb2,axiom,
    ! [Xa: int,A2: set_int] : insert_int(Xa,insert_int(Xa,A2)) = insert_int(Xa,A2) ).

% insert_absorb2
tff(fact_103_insert__absorb2,axiom,
    ! [Xa: $o,A2: set_o] : insert_o((Xa),insert_o((Xa),A2)) = insert_o((Xa),A2) ).

% insert_absorb2
tff(fact_104_insert__absorb2,axiom,
    ! [Xa: set_nat,A2: set_set_nat] : insert_set_nat(Xa,insert_set_nat(Xa,A2)) = insert_set_nat(Xa,A2) ).

% insert_absorb2
tff(fact_105_insert__absorb2,axiom,
    ! [Xa: real,A2: set_real] : insert_real(Xa,insert_real(Xa,A2)) = insert_real(Xa,A2) ).

% insert_absorb2
tff(fact_106_insert__absorb2,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(Xa,insert_Extended_enat(Xa,A2)) = insert_Extended_enat(Xa,A2) ).

% insert_absorb2
tff(fact_107_insert__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),insert_Extended_enat(B2,A2))
    <=> ( ( A = B2 )
        | aa_set3337841544853333262enat_o(member_Extended_enat(A),A2) ) ) ).

% insert_iff
tff(fact_108_insert__iff,axiom,
    ! [A: real,B2: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B2,A2))
    <=> ( ( A = B2 )
        | aa_set_real_o(member_real(A),A2) ) ) ).

% insert_iff
tff(fact_109_insert__iff,axiom,
    ! [A: set_nat,B2: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B2,A2))
    <=> ( ( A = B2 )
        | aa_set_set_nat_o(member_set_nat(A),A2) ) ) ).

% insert_iff
tff(fact_110_insert__iff,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B2,A2))
    <=> ( ( A = B2 )
        | aa_set_nat_o(member_nat(A),A2) ) ) ).

% insert_iff
tff(fact_111_insert__iff,axiom,
    ! [A: int,B2: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B2,A2))
    <=> ( ( A = B2 )
        | aa_set_int_o(member_int(A),A2) ) ) ).

% insert_iff
tff(fact_112_insert__iff,axiom,
    ! [A: $o,B2: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B2),A2))
    <=> ( ( (A)
        <=> (B2) )
        | aa_set_o_o(member_o((A)),A2) ) ) ).

% insert_iff
tff(fact_113_insertCI,axiom,
    ! [A: extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),B)
       => ( A = B2 ) )
     => aa_set3337841544853333262enat_o(member_Extended_enat(A),insert_Extended_enat(B2,B)) ) ).

% insertCI
tff(fact_114_insertCI,axiom,
    ! [A: real,B: set_real,B2: real] :
      ( ( ~ aa_set_real_o(member_real(A),B)
       => ( A = B2 ) )
     => aa_set_real_o(member_real(A),insert_real(B2,B)) ) ).

% insertCI
tff(fact_115_insertCI,axiom,
    ! [A: set_nat,B: set_set_nat,B2: set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(A),B)
       => ( A = B2 ) )
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B2,B)) ) ).

% insertCI
tff(fact_116_insertCI,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B)
       => ( A = B2 ) )
     => aa_set_nat_o(member_nat(A),insert_nat(B2,B)) ) ).

% insertCI
tff(fact_117_insertCI,axiom,
    ! [A: int,B: set_int,B2: int] :
      ( ( ~ aa_set_int_o(member_int(A),B)
       => ( A = B2 ) )
     => aa_set_int_o(member_int(A),insert_int(B2,B)) ) ).

% insertCI
tff(fact_118_insertCI,axiom,
    ! [A: $o,B: set_o,B2: $o] :
      ( ( ~ aa_set_o_o(member_o((A)),B)
       => ( (A)
        <=> (B2) ) )
     => aa_set_o_o(member_o((A)),insert_o((B2),B)) ) ).

% insertCI
tff(fact_119_sup_Oright__idem,axiom,
    ! [A: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) ).

% sup.right_idem
tff(fact_120_sup_Oright__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) ).

% sup.right_idem
tff(fact_121_sup_Oright__idem,axiom,
    ! [A: set_o,B2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)),B2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) ).

% sup.right_idem
tff(fact_122_sup_Oright__idem,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)),B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) ).

% sup.right_idem
tff(fact_123_sup_Oright__idem,axiom,
    ! [A: filter_nat,B2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)),B2) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) ).

% sup.right_idem
tff(fact_124_sup__left__idem,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) ).

% sup_left_idem
tff(fact_125_sup__left__idem,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) ).

% sup_left_idem
tff(fact_126_sup__left__idem,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) ).

% sup_left_idem
tff(fact_127_sup__left__idem,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) ).

% sup_left_idem
tff(fact_128_sup__left__idem,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) ).

% sup_left_idem
tff(fact_129_sup_Oleft__idem,axiom,
    ! [A: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) ).

% sup.left_idem
tff(fact_130_sup_Oleft__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) ).

% sup.left_idem
tff(fact_131_sup_Oleft__idem,axiom,
    ! [A: set_o,B2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) ).

% sup.left_idem
tff(fact_132_sup_Oleft__idem,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) ).

% sup.left_idem
tff(fact_133_sup_Oleft__idem,axiom,
    ! [A: filter_nat,B2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) ).

% sup.left_idem
tff(fact_134_sup__idem,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Xa) = Xa ).

% sup_idem
tff(fact_135_sup__idem,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Xa) = Xa ).

% sup_idem
tff(fact_136_sup__idem,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Xa) = Xa ).

% sup_idem
tff(fact_137_sup__idem,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Xa) = Xa ).

% sup_idem
tff(fact_138_sup__idem,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Xa) = Xa ).

% sup_idem
tff(fact_139_sup_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).

% sup.idem
tff(fact_140_sup_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),A) = A ).

% sup.idem
tff(fact_141_sup_Oidem,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),A) = A ).

% sup.idem
tff(fact_142_sup_Oidem,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),A) = A ).

% sup.idem
tff(fact_143_sup_Oidem,axiom,
    ! [A: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),A) = A ).

% sup.idem
tff(fact_144_Un__iff,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),sup_su4489774667511045786d_enat(A2,B))
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
        | aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% Un_iff
tff(fact_145_Un__iff,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),sup_sup_set_real(A2,B))
    <=> ( aa_set_real_o(member_real(C2),A2)
        | aa_set_real_o(member_real(C2),B) ) ) ).

% Un_iff
tff(fact_146_Un__iff,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A2,B))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A2)
        | aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% Un_iff
tff(fact_147_Un__iff,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))
    <=> ( aa_set_nat_o(member_nat(C2),A2)
        | aa_set_nat_o(member_nat(C2),B) ) ) ).

% Un_iff
tff(fact_148_Un__iff,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
    <=> ( aa_set_o_o(member_o((C2)),A2)
        | aa_set_o_o(member_o((C2)),B) ) ) ).

% Un_iff
tff(fact_149_Un__iff,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))
    <=> ( aa_set_int_o(member_int(C2),A2)
        | aa_set_int_o(member_int(C2),B) ) ) ).

% Un_iff
tff(fact_150_UnCI,axiom,
    ! [C2: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),B)
       => aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2) )
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),sup_su4489774667511045786d_enat(A2,B)) ) ).

% UnCI
tff(fact_151_UnCI,axiom,
    ! [C2: real,B: set_real,A2: set_real] :
      ( ( ~ aa_set_real_o(member_real(C2),B)
       => aa_set_real_o(member_real(C2),A2) )
     => aa_set_real_o(member_real(C2),sup_sup_set_real(A2,B)) ) ).

% UnCI
tff(fact_152_UnCI,axiom,
    ! [C2: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(C2),B)
       => aa_set_set_nat_o(member_set_nat(C2),A2) )
     => aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A2,B)) ) ).

% UnCI
tff(fact_153_UnCI,axiom,
    ! [C2: nat,B: set_nat,A2: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(C2),B)
       => aa_set_nat_o(member_nat(C2),A2) )
     => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) ) ).

% UnCI
tff(fact_154_UnCI,axiom,
    ! [C2: $o,B: set_o,A2: set_o] :
      ( ( ~ aa_set_o_o(member_o((C2)),B)
       => aa_set_o_o(member_o((C2)),A2) )
     => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) ) ).

% UnCI
tff(fact_155_UnCI,axiom,
    ! [C2: int,B: set_int,A2: set_int] :
      ( ( ~ aa_set_int_o(member_int(C2),B)
       => aa_set_int_o(member_int(C2),A2) )
     => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) ) ).

% UnCI
tff(fact_156_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_157_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_158_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),bot_bot_set_o) = A ).

% sup_bot.right_neutral
tff(fact_159_sup__bot_Oright__neutral,axiom,
    ! [A: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),bot_bot_filter_nat) = A ).

% sup_bot.right_neutral
tff(fact_160_sup__bot_Oright__neutral,axiom,
    ! [A: set_set_nat] : sup_sup_set_set_nat(A,bot_bot_set_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_161_sup__bot_Oright__neutral,axiom,
    ! [A: set_real] : sup_sup_set_real(A,bot_bot_set_real) = A ).

% sup_bot.right_neutral
tff(fact_162_sup__bot_Oright__neutral,axiom,
    ! [A: set_Extended_enat] : sup_su4489774667511045786d_enat(A,bot_bo7653980558646680370d_enat) = A ).

% sup_bot.right_neutral
tff(fact_163_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_164_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_165_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_166_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_167_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_168_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_169_bot__set__def,axiom,
    bot_bo7653980558646680370d_enat = collec4429806609662206161d_enat(bot_bo3575530754339543945enat_o) ).

% bot_set_def
tff(fact_170_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X: nat] : aa_set_nat_o(member_nat(X),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_171_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X: int] : aa_set_int_o(member_int(X),A2)
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_172_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ? [X: $o] : aa_set_o_o(member_o((X)),A2)
    <=> ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_173_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X: set_nat] : aa_set_set_nat_o(member_set_nat(X),A2)
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_174_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X: real] : aa_set_real_o(member_real(X),A2)
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_175_ex__in__conv,axiom,
    ! [A2: set_Extended_enat] :
      ( ? [X: extended_enat] : aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
    <=> ( A2 != bot_bo7653980558646680370d_enat ) ) ).

% ex_in_conv
tff(fact_176_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y: nat] : ~ aa_set_nat_o(member_nat(Y),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_177_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y: int] : ~ aa_set_int_o(member_int(Y),A2)
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_178_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y: $o] : ~ aa_set_o_o(member_o((Y)),A2)
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_179_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y),A2)
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_180_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y: real] : ~ aa_set_real_o(member_real(Y),A2)
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_181_equals0I,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [Y: extended_enat] : ~ aa_set3337841544853333262enat_o(member_Extended_enat(Y),A2)
     => ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% equals0I
tff(fact_182_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A2) ) ).

% equals0D
tff(fact_183_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A2) ) ).

% equals0D
tff(fact_184_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A2) ) ).

% equals0D
tff(fact_185_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A2) ) ).

% equals0D
tff(fact_186_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A),A2) ) ).

% equals0D
tff(fact_187_equals0D,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( A2 = bot_bo7653980558646680370d_enat )
     => ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),A2) ) ).

% equals0D
tff(fact_188_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_189_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_190_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_191_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_192_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real(A),bot_bot_set_real) ).

% emptyE
tff(fact_193_emptyE,axiom,
    ! [A: extended_enat] : ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),bot_bo7653980558646680370d_enat) ).

% emptyE
tff(fact_194_mk__disjoint__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ? [B3: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(A,B3) )
          & ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_195_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ? [B3: set_real] :
          ( ( A2 = insert_real(A,B3) )
          & ~ aa_set_real_o(member_real(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_196_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ? [B3: set_set_nat] :
          ( ( A2 = insert_set_nat(A,B3) )
          & ~ aa_set_set_nat_o(member_set_nat(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_197_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ? [B3: set_nat] :
          ( ( A2 = insert_nat(A,B3) )
          & ~ aa_set_nat_o(member_nat(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_198_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ? [B3: set_int] :
          ( ( A2 = insert_int(A,B3) )
          & ~ aa_set_int_o(member_int(A),B3) ) ) ).

% mk_disjoint_insert
tff(fact_199_mk__disjoint__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ? [B3: set_o] :
          ( ( A2 = insert_o((A),B3) )
          & ~ aa_set_o_o(member_o((A)),B3) ) ) ).

% mk_disjoint_insert
tff(fact_200_insert__commute,axiom,
    ! [Xa: nat,Ya: nat,A2: set_nat] : insert_nat(Xa,insert_nat(Ya,A2)) = insert_nat(Ya,insert_nat(Xa,A2)) ).

% insert_commute
tff(fact_201_insert__commute,axiom,
    ! [Xa: int,Ya: int,A2: set_int] : insert_int(Xa,insert_int(Ya,A2)) = insert_int(Ya,insert_int(Xa,A2)) ).

% insert_commute
tff(fact_202_insert__commute,axiom,
    ! [Xa: $o,Ya: $o,A2: set_o] : insert_o((Xa),insert_o((Ya),A2)) = insert_o((Ya),insert_o((Xa),A2)) ).

% insert_commute
tff(fact_203_insert__commute,axiom,
    ! [Xa: set_nat,Ya: set_nat,A2: set_set_nat] : insert_set_nat(Xa,insert_set_nat(Ya,A2)) = insert_set_nat(Ya,insert_set_nat(Xa,A2)) ).

% insert_commute
tff(fact_204_insert__commute,axiom,
    ! [Xa: real,Ya: real,A2: set_real] : insert_real(Xa,insert_real(Ya,A2)) = insert_real(Ya,insert_real(Xa,A2)) ).

% insert_commute
tff(fact_205_insert__commute,axiom,
    ! [Xa: extended_enat,Ya: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(Xa,insert_Extended_enat(Ya,A2)) = insert_Extended_enat(Ya,insert_Extended_enat(Xa,A2)) ).

% insert_commute
tff(fact_206_insert__eq__iff,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B2: extended_enat,B: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(B2),B)
       => ( ( insert_Extended_enat(A,A2) = insert_Extended_enat(B2,B) )
        <=> $ite(
              A = B2,
              A2 = B,
              ? [C3: set_Extended_enat] :
                ( ( A2 = insert_Extended_enat(B2,C3) )
                & ~ aa_set3337841544853333262enat_o(member_Extended_enat(B2),C3)
                & ( B = insert_Extended_enat(A,C3) )
                & ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_207_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B2: real,B: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( ~ aa_set_real_o(member_real(B2),B)
       => ( ( insert_real(A,A2) = insert_real(B2,B) )
        <=> $ite(
              A = B2,
              A2 = B,
              ? [C3: set_real] :
                ( ( A2 = insert_real(B2,C3) )
                & ~ aa_set_real_o(member_real(B2),C3)
                & ( B = insert_real(A,C3) )
                & ~ aa_set_real_o(member_real(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_208_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_nat,B: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(B2),B)
       => ( ( insert_set_nat(A,A2) = insert_set_nat(B2,B) )
        <=> $ite(
              A = B2,
              A2 = B,
              ? [C3: set_set_nat] :
                ( ( A2 = insert_set_nat(B2,C3) )
                & ~ aa_set_set_nat_o(member_set_nat(B2),C3)
                & ( B = insert_set_nat(A,C3) )
                & ~ aa_set_set_nat_o(member_set_nat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_209_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( ~ aa_set_nat_o(member_nat(B2),B)
       => ( ( insert_nat(A,A2) = insert_nat(B2,B) )
        <=> $ite(
              A = B2,
              A2 = B,
              ? [C3: set_nat] :
                ( ( A2 = insert_nat(B2,C3) )
                & ~ aa_set_nat_o(member_nat(B2),C3)
                & ( B = insert_nat(A,C3) )
                & ~ aa_set_nat_o(member_nat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_210_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B2: int,B: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( ~ aa_set_int_o(member_int(B2),B)
       => ( ( insert_int(A,A2) = insert_int(B2,B) )
        <=> $ite(
              A = B2,
              A2 = B,
              ? [C3: set_int] :
                ( ( A2 = insert_int(B2,C3) )
                & ~ aa_set_int_o(member_int(B2),C3)
                & ( B = insert_int(A,C3) )
                & ~ aa_set_int_o(member_int(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_211_insert__eq__iff,axiom,
    ! [A: $o,A2: set_o,B2: $o,B: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( ~ aa_set_o_o(member_o((B2)),B)
       => ( ( insert_o((A),A2) = insert_o((B2),B) )
        <=> $ite(
              (A) = (B2),
              A2 = B,
              ? [C3: set_o] :
                ( ( A2 = insert_o((B2),C3) )
                & ~ aa_set_o_o(member_o((B2)),C3)
                & ( B = insert_o((A),C3) )
                & ~ aa_set_o_o(member_o((A)),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_212_insert__absorb,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ( insert_Extended_enat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_213_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_214_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_215_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_216_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_217_insert__absorb,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),A2) = A2 ) ) ).

% insert_absorb
tff(fact_218_insert__ident,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B)
       => ( ( insert_Extended_enat(Xa,A2) = insert_Extended_enat(Xa,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_219_insert__ident,axiom,
    ! [Xa: real,A2: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( ~ aa_set_real_o(member_real(Xa),B)
       => ( ( insert_real(Xa,A2) = insert_real(Xa,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_220_insert__ident,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),B)
       => ( ( insert_set_nat(Xa,A2) = insert_set_nat(Xa,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_221_insert__ident,axiom,
    ! [Xa: nat,A2: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),B)
       => ( ( insert_nat(Xa,A2) = insert_nat(Xa,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_222_insert__ident,axiom,
    ! [Xa: int,A2: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( ~ aa_set_int_o(member_int(Xa),B)
       => ( ( insert_int(Xa,A2) = insert_int(Xa,B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_223_insert__ident,axiom,
    ! [Xa: $o,A2: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( ~ aa_set_o_o(member_o((Xa)),B)
       => ( ( insert_o((Xa),A2) = insert_o((Xa),B) )
        <=> ( A2 = B ) ) ) ) ).

% insert_ident
tff(fact_224_Set_Oset__insert,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ~ ! [B3: set_Extended_enat] :
            ( ( A2 = insert_Extended_enat(Xa,B3) )
           => aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_225_Set_Oset__insert,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),A2)
     => ~ ! [B3: set_real] :
            ( ( A2 = insert_real(Xa,B3) )
           => aa_set_real_o(member_real(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_226_Set_Oset__insert,axiom,
    ! [Xa: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ~ ! [B3: set_set_nat] :
            ( ( A2 = insert_set_nat(Xa,B3) )
           => aa_set_set_nat_o(member_set_nat(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_227_Set_Oset__insert,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A2)
     => ~ ! [B3: set_nat] :
            ( ( A2 = insert_nat(Xa,B3) )
           => aa_set_nat_o(member_nat(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_228_Set_Oset__insert,axiom,
    ! [Xa: int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),A2)
     => ~ ! [B3: set_int] :
            ( ( A2 = insert_int(Xa,B3) )
           => aa_set_int_o(member_int(Xa),B3) ) ) ).

% Set.set_insert
tff(fact_229_Set_Oset__insert,axiom,
    ! [Xa: $o,A2: set_o] :
      ( aa_set_o_o(member_o((Xa)),A2)
     => ~ ! [B3: set_o] :
            ( ( A2 = insert_o((Xa),B3) )
           => aa_set_o_o(member_o((Xa)),B3) ) ) ).

% Set.set_insert
tff(fact_230_insertI2,axiom,
    ! [A: extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),B)
     => aa_set3337841544853333262enat_o(member_Extended_enat(A),insert_Extended_enat(B2,B)) ) ).

% insertI2
tff(fact_231_insertI2,axiom,
    ! [A: real,B: set_real,B2: real] :
      ( aa_set_real_o(member_real(A),B)
     => aa_set_real_o(member_real(A),insert_real(B2,B)) ) ).

% insertI2
tff(fact_232_insertI2,axiom,
    ! [A: set_nat,B: set_set_nat,B2: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),B)
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B2,B)) ) ).

% insertI2
tff(fact_233_insertI2,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( aa_set_nat_o(member_nat(A),B)
     => aa_set_nat_o(member_nat(A),insert_nat(B2,B)) ) ).

% insertI2
tff(fact_234_insertI2,axiom,
    ! [A: int,B: set_int,B2: int] :
      ( aa_set_int_o(member_int(A),B)
     => aa_set_int_o(member_int(A),insert_int(B2,B)) ) ).

% insertI2
tff(fact_235_insertI2,axiom,
    ! [A: $o,B: set_o,B2: $o] :
      ( aa_set_o_o(member_o((A)),B)
     => aa_set_o_o(member_o((A)),insert_o((B2),B)) ) ).

% insertI2
tff(fact_236_insertI1,axiom,
    ! [A: extended_enat,B: set_Extended_enat] : aa_set3337841544853333262enat_o(member_Extended_enat(A),insert_Extended_enat(A,B)) ).

% insertI1
tff(fact_237_insertI1,axiom,
    ! [A: real,B: set_real] : aa_set_real_o(member_real(A),insert_real(A,B)) ).

% insertI1
tff(fact_238_insertI1,axiom,
    ! [A: set_nat,B: set_set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,B)) ).

% insertI1
tff(fact_239_insertI1,axiom,
    ! [A: nat,B: set_nat] : aa_set_nat_o(member_nat(A),insert_nat(A,B)) ).

% insertI1
tff(fact_240_insertI1,axiom,
    ! [A: int,B: set_int] : aa_set_int_o(member_int(A),insert_int(A,B)) ).

% insertI1
tff(fact_241_insertI1,axiom,
    ! [A: $o,B: set_o] : aa_set_o_o(member_o((A)),insert_o((A),B)) ).

% insertI1
tff(fact_242_insertE,axiom,
    ! [A: extended_enat,B2: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),insert_Extended_enat(B2,A2))
     => ( ( A != B2 )
       => aa_set3337841544853333262enat_o(member_Extended_enat(A),A2) ) ) ).

% insertE
tff(fact_243_insertE,axiom,
    ! [A: real,B2: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B2,A2))
     => ( ( A != B2 )
       => aa_set_real_o(member_real(A),A2) ) ) ).

% insertE
tff(fact_244_insertE,axiom,
    ! [A: set_nat,B2: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B2,A2))
     => ( ( A != B2 )
       => aa_set_set_nat_o(member_set_nat(A),A2) ) ) ).

% insertE
tff(fact_245_insertE,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B2,A2))
     => ( ( A != B2 )
       => aa_set_nat_o(member_nat(A),A2) ) ) ).

% insertE
tff(fact_246_insertE,axiom,
    ! [A: int,B2: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B2,A2))
     => ( ( A != B2 )
       => aa_set_int_o(member_int(A),A2) ) ) ).

% insertE
tff(fact_247_insertE,axiom,
    ! [A: $o,B2: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B2),A2))
     => ( ( (A)
        <=> ~ (B2) )
       => aa_set_o_o(member_o((A)),A2) ) ) ).

% insertE
tff(fact_248_sup__left__commute,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% sup_left_commute
tff(fact_249_sup__left__commute,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ).

% sup_left_commute
tff(fact_250_sup__left__commute,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Z)) ).

% sup_left_commute
tff(fact_251_sup__left__commute,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Z)) ).

% sup_left_commute
tff(fact_252_sup__left__commute,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Z)) ).

% sup_left_commute
tff(fact_253_mem__Collect__eq,axiom,
    ! [A: extended_enat,P: fun_Extended_enat_o] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),collec4429806609662206161d_enat(P))
    <=> aa_Extended_enat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_254_mem__Collect__eq,axiom,
    ! [A: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o((A)),collect_o(P))
    <=> aa_o_o(P,(A)) ) ).

% mem_Collect_eq
tff(fact_255_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A),collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_256_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> aa_list_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_257_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_258_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_259_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_260_Collect__mem__eq,axiom,
    ! [A2: set_Extended_enat] : collec4429806609662206161d_enat(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_261_Collect__mem__eq,axiom,
    ! [A2: set_o] : collect_o(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_262_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_263_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_264_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_265_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_266_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_af(A2)) = A2 ).

% Collect_mem_eq
tff(fact_267_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X2: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(Q,X2) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_268_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X2: list_nat] :
          ( aa_list_nat_o(P,X2)
        <=> aa_list_nat_o(Q,X2) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_269_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
        <=> aa_set_nat_o(Q,X2) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_270_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_271_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_272_sup_Oleft__commute,axiom,
    ! [B2: set_nat,A: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% sup.left_commute
tff(fact_273_sup_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C2)) ).

% sup.left_commute
tff(fact_274_sup_Oleft__commute,axiom,
    ! [B2: set_o,A: set_o,C2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),C2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)) ).

% sup.left_commute
tff(fact_275_sup_Oleft__commute,axiom,
    ! [B2: set_int,A: set_int,C2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)) ).

% sup.left_commute
tff(fact_276_sup_Oleft__commute,axiom,
    ! [B2: filter_nat,A: filter_nat,C2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),C2)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C2)) ).

% sup.left_commute
tff(fact_277_sup__commute,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Xa) ).

% sup_commute
tff(fact_278_sup__commute,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Xa) ).

% sup_commute
tff(fact_279_sup__commute,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Xa) ).

% sup_commute
tff(fact_280_sup__commute,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Xa) ).

% sup_commute
tff(fact_281_sup__commute,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Xa) ).

% sup_commute
tff(fact_282_sup_Ocommute,axiom,
    ! [A: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A) ).

% sup.commute
tff(fact_283_sup_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),A) ).

% sup.commute
tff(fact_284_sup_Ocommute,axiom,
    ! [A: set_o,B2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),A) ).

% sup.commute
tff(fact_285_sup_Ocommute,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),A) ).

% sup.commute
tff(fact_286_sup_Ocommute,axiom,
    ! [A: filter_nat,B2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),A) ).

% sup.commute
tff(fact_287_sup__assoc,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) ).

% sup_assoc
tff(fact_288_sup__assoc,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)) ).

% sup_assoc
tff(fact_289_sup__assoc,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),Z) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) ).

% sup_assoc
tff(fact_290_sup__assoc,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),Z) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) ).

% sup_assoc
tff(fact_291_sup__assoc,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)),Z) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)) ).

% sup_assoc
tff(fact_292_sup_Oassoc,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% sup.assoc
tff(fact_293_sup_Oassoc,axiom,
    ! [A: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C2)) ).

% sup.assoc
tff(fact_294_sup_Oassoc,axiom,
    ! [A: set_o,B2: set_o,C2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)),C2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)) ).

% sup.assoc
tff(fact_295_sup_Oassoc,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)) ).

% sup.assoc
tff(fact_296_sup_Oassoc,axiom,
    ! [A: filter_nat,B2: filter_nat,C2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)),C2) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C2)) ).

% sup.assoc
tff(fact_297_inf__sup__aci_I5_J,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Xa) ).

% inf_sup_aci(5)
tff(fact_298_inf__sup__aci_I5_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Xa) ).

% inf_sup_aci(5)
tff(fact_299_inf__sup__aci_I5_J,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Xa) ).

% inf_sup_aci(5)
tff(fact_300_inf__sup__aci_I5_J,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Xa) ).

% inf_sup_aci(5)
tff(fact_301_inf__sup__aci_I5_J,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Xa) ).

% inf_sup_aci(5)
tff(fact_302_inf__sup__aci_I6_J,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) ).

% inf_sup_aci(6)
tff(fact_303_inf__sup__aci_I6_J,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)) ).

% inf_sup_aci(6)
tff(fact_304_inf__sup__aci_I6_J,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),Z) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) ).

% inf_sup_aci(6)
tff(fact_305_inf__sup__aci_I6_J,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),Z) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) ).

% inf_sup_aci(6)
tff(fact_306_inf__sup__aci_I6_J,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)),Z) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)) ).

% inf_sup_aci(6)
tff(fact_307_inf__sup__aci_I7_J,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_308_inf__sup__aci_I7_J,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_309_inf__sup__aci_I7_J,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_310_inf__sup__aci_I7_J,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_311_inf__sup__aci_I7_J,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Z)) ).

% inf_sup_aci(7)
tff(fact_312_inf__sup__aci_I8_J,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) ).

% inf_sup_aci(8)
tff(fact_313_inf__sup__aci_I8_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) ).

% inf_sup_aci(8)
tff(fact_314_inf__sup__aci_I8_J,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) ).

% inf_sup_aci(8)
tff(fact_315_inf__sup__aci_I8_J,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) ).

% inf_sup_aci(8)
tff(fact_316_inf__sup__aci_I8_J,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) ).

% inf_sup_aci(8)
tff(fact_317_Un__left__commute,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C)) ).

% Un_left_commute
tff(fact_318_Un__left__commute,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),C)) ).

% Un_left_commute
tff(fact_319_Un__left__commute,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),C)) ).

% Un_left_commute
tff(fact_320_Un__left__absorb,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) ).

% Un_left_absorb
tff(fact_321_Un__left__absorb,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) ).

% Un_left_absorb
tff(fact_322_Un__left__absorb,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) ).

% Un_left_absorb
tff(fact_323_Un__commute,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A2) ).

% Un_commute
tff(fact_324_Un__commute,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),A2) ).

% Un_commute
tff(fact_325_Un__commute,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),A2) ).

% Un_commute
tff(fact_326_Un__absorb,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),A2) = A2 ).

% Un_absorb
tff(fact_327_Un__absorb,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),A2) = A2 ).

% Un_absorb
tff(fact_328_Un__absorb,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),A2) = A2 ).

% Un_absorb
tff(fact_329_Un__assoc,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).

% Un_assoc
tff(fact_330_Un__assoc,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),C) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) ).

% Un_assoc
tff(fact_331_Un__assoc,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),C) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) ).

% Un_assoc
tff(fact_332_ball__Un,axiom,
    ! [A2: set_nat,B: set_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))
         => aa_nat_o(P,X) )
    <=> ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
           => aa_nat_o(P,X) )
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B)
           => aa_nat_o(P,X) ) ) ) ).

% ball_Un
tff(fact_333_ball__Un,axiom,
    ! [A2: set_o,B: set_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
         => aa_o_o(P,(X)) )
    <=> ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A2)
           => aa_o_o(P,(X)) )
        & ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B)
           => aa_o_o(P,(X)) ) ) ) ).

% ball_Un
tff(fact_334_ball__Un,axiom,
    ! [A2: set_int,B: set_int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))
         => aa_int_o(P,X) )
    <=> ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A2)
           => aa_int_o(P,X) )
        & ! [X: int] :
            ( aa_set_int_o(member_int(X),B)
           => aa_int_o(P,X) ) ) ) ).

% ball_Un
tff(fact_335_bex__Un,axiom,
    ! [A2: set_nat,B: set_nat,P: fun_nat_o] :
      ( ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))
          & aa_nat_o(P,X) )
    <=> ( ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
            & aa_nat_o(P,X) )
        | ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),B)
            & aa_nat_o(P,X) ) ) ) ).

% bex_Un
tff(fact_336_bex__Un,axiom,
    ! [A2: set_o,B: set_o,P: fun_o_o] :
      ( ? [X: $o] :
          ( aa_set_o_o(member_o((X)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
          & aa_o_o(P,(X)) )
    <=> ( ? [X: $o] :
            ( aa_set_o_o(member_o((X)),A2)
            & aa_o_o(P,(X)) )
        | ? [X: $o] :
            ( aa_set_o_o(member_o((X)),B)
            & aa_o_o(P,(X)) ) ) ) ).

% bex_Un
tff(fact_337_bex__Un,axiom,
    ! [A2: set_int,B: set_int,P: fun_int_o] :
      ( ? [X: int] :
          ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))
          & aa_int_o(P,X) )
    <=> ( ? [X: int] :
            ( aa_set_int_o(member_int(X),A2)
            & aa_int_o(P,X) )
        | ? [X: int] :
            ( aa_set_int_o(member_int(X),B)
            & aa_int_o(P,X) ) ) ) ).

% bex_Un
tff(fact_338_UnI2,axiom,
    ! [C2: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),B)
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),sup_su4489774667511045786d_enat(A2,B)) ) ).

% UnI2
tff(fact_339_UnI2,axiom,
    ! [C2: real,B: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(C2),B)
     => aa_set_real_o(member_real(C2),sup_sup_set_real(A2,B)) ) ).

% UnI2
tff(fact_340_UnI2,axiom,
    ! [C2: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),B)
     => aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A2,B)) ) ).

% UnI2
tff(fact_341_UnI2,axiom,
    ! [C2: nat,B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C2),B)
     => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) ) ).

% UnI2
tff(fact_342_UnI2,axiom,
    ! [C2: $o,B: set_o,A2: set_o] :
      ( aa_set_o_o(member_o((C2)),B)
     => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) ) ).

% UnI2
tff(fact_343_UnI2,axiom,
    ! [C2: int,B: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(C2),B)
     => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) ) ).

% UnI2
tff(fact_344_UnI1,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),sup_su4489774667511045786d_enat(A2,B)) ) ).

% UnI1
tff(fact_345_UnI1,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),A2)
     => aa_set_real_o(member_real(C2),sup_sup_set_real(A2,B)) ) ).

% UnI1
tff(fact_346_UnI1,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A2)
     => aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A2,B)) ) ).

% UnI1
tff(fact_347_UnI1,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A2)
     => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) ) ).

% UnI1
tff(fact_348_UnI1,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),A2)
     => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) ) ).

% UnI1
tff(fact_349_UnI1,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),A2)
     => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) ) ).

% UnI1
tff(fact_350_UnE,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),sup_su4489774667511045786d_enat(A2,B))
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
       => aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% UnE
tff(fact_351_UnE,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),sup_sup_set_real(A2,B))
     => ( ~ aa_set_real_o(member_real(C2),A2)
       => aa_set_real_o(member_real(C2),B) ) ) ).

% UnE
tff(fact_352_UnE,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A2,B))
     => ( ~ aa_set_set_nat_o(member_set_nat(C2),A2)
       => aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% UnE
tff(fact_353_UnE,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))
     => ( ~ aa_set_nat_o(member_nat(C2),A2)
       => aa_set_nat_o(member_nat(C2),B) ) ) ).

% UnE
tff(fact_354_UnE,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
     => ( ~ aa_set_o_o(member_o((C2)),A2)
       => aa_set_o_o(member_o((C2)),B) ) ) ).

% UnE
tff(fact_355_UnE,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))
     => ( ~ aa_set_int_o(member_int(C2),A2)
       => aa_set_int_o(member_int(C2),B) ) ) ).

% UnE
tff(fact_356_singleton__inject,axiom,
    ! [A: nat,B2: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B2,bot_bot_set_nat) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_357_singleton__inject,axiom,
    ! [A: int,B2: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B2,bot_bot_set_int) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_358_singleton__inject,axiom,
    ! [A: $o,B2: $o] :
      ( ( insert_o((A),bot_bot_set_o) = insert_o((B2),bot_bot_set_o) )
     => ( (A)
      <=> (B2) ) ) ).

% singleton_inject
tff(fact_359_singleton__inject,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( insert_set_nat(A,bot_bot_set_set_nat) = insert_set_nat(B2,bot_bot_set_set_nat) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_360_singleton__inject,axiom,
    ! [A: real,B2: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B2,bot_bot_set_real) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_361_singleton__inject,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( insert_Extended_enat(A,bot_bo7653980558646680370d_enat) = insert_Extended_enat(B2,bot_bo7653980558646680370d_enat) )
     => ( A = B2 ) ) ).

% singleton_inject
tff(fact_362_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_363_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_364_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),A2) != bot_bot_set_o ).

% insert_not_empty
tff(fact_365_insert__not__empty,axiom,
    ! [A: set_nat,A2: set_set_nat] : insert_set_nat(A,A2) != bot_bot_set_set_nat ).

% insert_not_empty
tff(fact_366_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_367_insert__not__empty,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(A,A2) != bot_bo7653980558646680370d_enat ).

% insert_not_empty
tff(fact_368_doubleton__eq__iff,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B2,bot_bot_set_nat)) = insert_nat(C2,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_369_doubleton__eq__iff,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( ( insert_int(A,insert_int(B2,bot_bot_set_int)) = insert_int(C2,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_370_doubleton__eq__iff,axiom,
    ! [A: $o,B2: $o,C2: $o,D: $o] :
      ( ( insert_o((A),insert_o((B2),bot_bot_set_o)) = insert_o((C2),insert_o((D),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C2) )
          & ( (B2)
          <=> (D) ) )
        | ( ( (A)
          <=> (D) )
          & ( (B2)
          <=> (C2) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_371_doubleton__eq__iff,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat,D: set_nat] :
      ( ( insert_set_nat(A,insert_set_nat(B2,bot_bot_set_set_nat)) = insert_set_nat(C2,insert_set_nat(D,bot_bot_set_set_nat)) )
    <=> ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_372_doubleton__eq__iff,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( ( insert_real(A,insert_real(B2,bot_bot_set_real)) = insert_real(C2,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_373_doubleton__eq__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C2: extended_enat,D: extended_enat] :
      ( ( insert_Extended_enat(A,insert_Extended_enat(B2,bot_bo7653980558646680370d_enat)) = insert_Extended_enat(C2,insert_Extended_enat(D,bot_bo7653980558646680370d_enat)) )
    <=> ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_374_singleton__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B2),insert_nat(A,bot_bot_set_nat))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_375_singleton__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_set_int_o(member_int(B2),insert_int(A,bot_bot_set_int))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_376_singleton__iff,axiom,
    ! [B2: $o,A: $o] :
      ( aa_set_o_o(member_o((B2)),insert_o((A),bot_bot_set_o))
    <=> ( (B2)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_377_singleton__iff,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B2),insert_set_nat(A,bot_bot_set_set_nat))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_378_singleton__iff,axiom,
    ! [B2: real,A: real] :
      ( aa_set_real_o(member_real(B2),insert_real(A,bot_bot_set_real))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_379_singleton__iff,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(B2),insert_Extended_enat(A,bot_bo7653980558646680370d_enat))
    <=> ( B2 = A ) ) ).

% singleton_iff
tff(fact_380_singletonD,axiom,
    ! [B2: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B2),insert_nat(A,bot_bot_set_nat))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_381_singletonD,axiom,
    ! [B2: int,A: int] :
      ( aa_set_int_o(member_int(B2),insert_int(A,bot_bot_set_int))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_382_singletonD,axiom,
    ! [B2: $o,A: $o] :
      ( aa_set_o_o(member_o((B2)),insert_o((A),bot_bot_set_o))
     => ( (B2)
      <=> (A) ) ) ).

% singletonD
tff(fact_383_singletonD,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B2),insert_set_nat(A,bot_bot_set_set_nat))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_384_singletonD,axiom,
    ! [B2: real,A: real] :
      ( aa_set_real_o(member_real(B2),insert_real(A,bot_bot_set_real))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_385_singletonD,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(B2),insert_Extended_enat(A,bot_bo7653980558646680370d_enat))
     => ( B2 = A ) ) ).

% singletonD
tff(fact_386_Un__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ).

% Un_empty_right
tff(fact_387_Un__empty__right,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),bot_bot_set_int) = A2 ).

% Un_empty_right
tff(fact_388_Un__empty__right,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),bot_bot_set_o) = A2 ).

% Un_empty_right
tff(fact_389_Un__empty__right,axiom,
    ! [A2: set_set_nat] : sup_sup_set_set_nat(A2,bot_bot_set_set_nat) = A2 ).

% Un_empty_right
tff(fact_390_Un__empty__right,axiom,
    ! [A2: set_real] : sup_sup_set_real(A2,bot_bot_set_real) = A2 ).

% Un_empty_right
tff(fact_391_Un__empty__right,axiom,
    ! [A2: set_Extended_enat] : sup_su4489774667511045786d_enat(A2,bot_bo7653980558646680370d_enat) = A2 ).

% Un_empty_right
tff(fact_392_Un__empty__left,axiom,
    ! [B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B) = B ).

% Un_empty_left
tff(fact_393_Un__empty__left,axiom,
    ! [B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B) = B ).

% Un_empty_left
tff(fact_394_Un__empty__left,axiom,
    ! [B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),B) = B ).

% Un_empty_left
tff(fact_395_Un__empty__left,axiom,
    ! [B: set_set_nat] : sup_sup_set_set_nat(bot_bot_set_set_nat,B) = B ).

% Un_empty_left
tff(fact_396_Un__empty__left,axiom,
    ! [B: set_real] : sup_sup_set_real(bot_bot_set_real,B) = B ).

% Un_empty_left
tff(fact_397_Un__empty__left,axiom,
    ! [B: set_Extended_enat] : sup_su4489774667511045786d_enat(bot_bo7653980558646680370d_enat,B) = B ).

% Un_empty_left
tff(fact_398_singleton__Un__iff,axiom,
    ! [Xa: nat,A2: set_nat,B: set_nat] :
      ( ( insert_nat(Xa,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B = insert_nat(Xa,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B = insert_nat(Xa,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_399_singleton__Un__iff,axiom,
    ! [Xa: int,A2: set_int,B: set_int] :
      ( ( insert_int(Xa,bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B = insert_int(Xa,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B = bot_bot_set_int ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B = insert_int(Xa,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_400_singleton__Un__iff,axiom,
    ! [Xa: $o,A2: set_o,B: set_o] :
      ( ( insert_o((Xa),bot_bot_set_o) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B = insert_o((Xa),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B = bot_bot_set_o ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B = insert_o((Xa),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_401_singleton__Un__iff,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( insert_set_nat(Xa,bot_bot_set_set_nat) = sup_sup_set_set_nat(A2,B) )
    <=> ( ( ( A2 = bot_bot_set_set_nat )
          & ( B = insert_set_nat(Xa,bot_bot_set_set_nat) ) )
        | ( ( A2 = insert_set_nat(Xa,bot_bot_set_set_nat) )
          & ( B = bot_bot_set_set_nat ) )
        | ( ( A2 = insert_set_nat(Xa,bot_bot_set_set_nat) )
          & ( B = insert_set_nat(Xa,bot_bot_set_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_402_singleton__Un__iff,axiom,
    ! [Xa: real,A2: set_real,B: set_real] :
      ( ( insert_real(Xa,bot_bot_set_real) = sup_sup_set_real(A2,B) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B = insert_real(Xa,bot_bot_set_real) ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B = bot_bot_set_real ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B = insert_real(Xa,bot_bot_set_real) ) ) ) ) ).

% singleton_Un_iff
tff(fact_403_singleton__Un__iff,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) = sup_su4489774667511045786d_enat(A2,B) )
    <=> ( ( ( A2 = bot_bo7653980558646680370d_enat )
          & ( B = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) ) )
        | ( ( A2 = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) )
          & ( B = bot_bo7653980558646680370d_enat ) )
        | ( ( A2 = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) )
          & ( B = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_404_Un__singleton__iff,axiom,
    ! [A2: set_nat,B: set_nat,Xa: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = insert_nat(Xa,bot_bot_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B = insert_nat(Xa,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B = insert_nat(Xa,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_405_Un__singleton__iff,axiom,
    ! [A2: set_int,B: set_int,Xa: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = insert_int(Xa,bot_bot_set_int) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B = insert_int(Xa,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B = bot_bot_set_int ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B = insert_int(Xa,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_406_Un__singleton__iff,axiom,
    ! [A2: set_o,B: set_o,Xa: $o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = insert_o((Xa),bot_bot_set_o) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B = insert_o((Xa),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B = bot_bot_set_o ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B = insert_o((Xa),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_407_Un__singleton__iff,axiom,
    ! [A2: set_set_nat,B: set_set_nat,Xa: set_nat] :
      ( ( sup_sup_set_set_nat(A2,B) = insert_set_nat(Xa,bot_bot_set_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_set_nat )
          & ( B = insert_set_nat(Xa,bot_bot_set_set_nat) ) )
        | ( ( A2 = insert_set_nat(Xa,bot_bot_set_set_nat) )
          & ( B = bot_bot_set_set_nat ) )
        | ( ( A2 = insert_set_nat(Xa,bot_bot_set_set_nat) )
          & ( B = insert_set_nat(Xa,bot_bot_set_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_408_Un__singleton__iff,axiom,
    ! [A2: set_real,B: set_real,Xa: real] :
      ( ( sup_sup_set_real(A2,B) = insert_real(Xa,bot_bot_set_real) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B = insert_real(Xa,bot_bot_set_real) ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B = bot_bot_set_real ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B = insert_real(Xa,bot_bot_set_real) ) ) ) ) ).

% Un_singleton_iff
tff(fact_409_Un__singleton__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,Xa: extended_enat] :
      ( ( sup_su4489774667511045786d_enat(A2,B) = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) )
    <=> ( ( ( A2 = bot_bo7653980558646680370d_enat )
          & ( B = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) ) )
        | ( ( A2 = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) )
          & ( B = bot_bo7653980558646680370d_enat ) )
        | ( ( A2 = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) )
          & ( B = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_410_insert__is__Un,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,bot_bot_set_nat)),A2) ).

% insert_is_Un
tff(fact_411_insert__is__Un,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A,bot_bot_set_int)),A2) ).

% insert_is_Un
tff(fact_412_insert__is__Un,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),A2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,insert_o((A),bot_bot_set_o)),A2) ).

% insert_is_Un
tff(fact_413_insert__is__Un,axiom,
    ! [A: set_nat,A2: set_set_nat] : insert_set_nat(A,A2) = sup_sup_set_set_nat(insert_set_nat(A,bot_bot_set_set_nat),A2) ).

% insert_is_Un
tff(fact_414_insert__is__Un,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) = sup_sup_set_real(insert_real(A,bot_bot_set_real),A2) ).

% insert_is_Un
tff(fact_415_insert__is__Un,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(A,A2) = sup_su4489774667511045786d_enat(insert_Extended_enat(A,bot_bo7653980558646680370d_enat),A2) ).

% insert_is_Un
tff(fact_416__092_060open_062set__vebt_H_At_A_092_060union_062_A_123x_125_A_092_060subseteq_062_Aset__vebt_H_A_Ivebt__insert_At_Ax_J_092_060close_062,axiom,
    aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_VEBT_set_vebt(t)),insert_nat(x,bot_bot_set_nat))),vEBT_VEBT_set_vebt(vEBT_vebt_insert(t,x))) ).

% \<open>set_vebt' t \<union> {x} \<subseteq> set_vebt' (vebt_insert t x)\<close>
tff(fact_417_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xa: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na)),Xa) ).

% buildup_nothing_in_leaf
tff(fact_418_assms_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(t),n) ).

% assms(1)
tff(fact_419_the__elem__eq,axiom,
    ! [Xa: nat] : the_elem_nat(insert_nat(Xa,bot_bot_set_nat)) = Xa ).

% the_elem_eq
tff(fact_420_the__elem__eq,axiom,
    ! [Xa: int] : the_elem_int(insert_int(Xa,bot_bot_set_int)) = Xa ).

% the_elem_eq
tff(fact_421_the__elem__eq,axiom,
    ! [Xa: $o] :
      ( aa_set_o_o(the_elem_o,insert_o((Xa),bot_bot_set_o))
    <=> (Xa) ) ).

% the_elem_eq
tff(fact_422_the__elem__eq,axiom,
    ! [Xa: set_nat] : the_elem_set_nat(insert_set_nat(Xa,bot_bot_set_set_nat)) = Xa ).

% the_elem_eq
tff(fact_423_the__elem__eq,axiom,
    ! [Xa: real] : the_elem_real(insert_real(Xa,bot_bot_set_real)) = Xa ).

% the_elem_eq
tff(fact_424_the__elem__eq,axiom,
    ! [Xa: extended_enat] : the_el319773668273709403d_enat(insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = Xa ).

% the_elem_eq
tff(fact_425_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xa: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Na)),Xa) ).

% buildup_nothing_in_min_max
tff(fact_426_is__singletonI,axiom,
    ! [Xa: nat] : aa_set_nat_o(is_singleton_nat,insert_nat(Xa,bot_bot_set_nat)) ).

% is_singletonI
tff(fact_427_is__singletonI,axiom,
    ! [Xa: int] : aa_set_int_o(is_singleton_int,insert_int(Xa,bot_bot_set_int)) ).

% is_singletonI
tff(fact_428_is__singletonI,axiom,
    ! [Xa: $o] : aa_set_o_o(is_singleton_o,insert_o((Xa),bot_bot_set_o)) ).

% is_singletonI
tff(fact_429_is__singletonI,axiom,
    ! [Xa: set_nat] : aa_set_set_nat_o(is_singleton_set_nat,insert_set_nat(Xa,bot_bot_set_set_nat)) ).

% is_singletonI
tff(fact_430_is__singletonI,axiom,
    ! [Xa: real] : aa_set_real_o(is_singleton_real,insert_real(Xa,bot_bot_set_real)) ).

% is_singletonI
tff(fact_431_is__singletonI,axiom,
    ! [Xa: extended_enat] : aa_set3337841544853333262enat_o(is_sin1871519699599484762d_enat,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) ).

% is_singletonI
tff(fact_432_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),bot_bot_set_nat) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_433_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),bot_bot_set_int) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_434_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),bot_bot_set_o) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_435_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_set_nat] : sup_sup_set_set_nat(Xa,bot_bot_set_set_nat) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_436_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_real] : sup_sup_set_real(Xa,bot_bot_set_real) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_437_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_Extended_enat] : sup_su4489774667511045786d_enat(Xa,bot_bo7653980558646680370d_enat) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_438_Set_Ois__empty__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_empty_nat,A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_439_Set_Ois__empty__def,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_empty_int,A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_440_Set_Ois__empty__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_empty_o,A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% Set.is_empty_def
tff(fact_441_Set_Ois__empty__def,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(is_empty_set_nat,A2)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% Set.is_empty_def
tff(fact_442_Set_Ois__empty__def,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_empty_real,A2)
    <=> ( A2 = bot_bot_set_real ) ) ).

% Set.is_empty_def
tff(fact_443_Set_Ois__empty__def,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(is_emp5240238520263478072d_enat,A2)
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% Set.is_empty_def
tff(fact_444_is__singleton__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ? [X: nat] : A2 = insert_nat(X,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_445_is__singleton__def,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
    <=> ? [X: int] : A2 = insert_int(X,bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_446_is__singleton__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ? [X: $o] : A2 = insert_o((X),bot_bot_set_o) ) ).

% is_singleton_def
tff(fact_447_is__singleton__def,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(is_singleton_set_nat,A2)
    <=> ? [X: set_nat] : A2 = insert_set_nat(X,bot_bot_set_set_nat) ) ).

% is_singleton_def
tff(fact_448_is__singleton__def,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_singleton_real,A2)
    <=> ? [X: real] : A2 = insert_real(X,bot_bot_set_real) ) ).

% is_singleton_def
tff(fact_449_is__singleton__def,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(is_sin1871519699599484762d_enat,A2)
    <=> ? [X: extended_enat] : A2 = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ).

% is_singleton_def
tff(fact_450_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
     => ~ ! [X2: nat] : A2 != insert_nat(X2,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_451_is__singletonE,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
     => ~ ! [X2: int] : A2 != insert_int(X2,bot_bot_set_int) ) ).

% is_singletonE
tff(fact_452_is__singletonE,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
     => ~ ! [X2: $o] : A2 != insert_o((X2),bot_bot_set_o) ) ).

% is_singletonE
tff(fact_453_is__singletonE,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(is_singleton_set_nat,A2)
     => ~ ! [X2: set_nat] : A2 != insert_set_nat(X2,bot_bot_set_set_nat) ) ).

% is_singletonE
tff(fact_454_is__singletonE,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_singleton_real,A2)
     => ~ ! [X2: real] : A2 != insert_real(X2,bot_bot_set_real) ) ).

% is_singletonE
tff(fact_455_is__singletonE,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(is_sin1871519699599484762d_enat,A2)
     => ~ ! [X2: extended_enat] : A2 != insert_Extended_enat(X2,bot_bo7653980558646680370d_enat) ) ).

% is_singletonE
tff(fact_456_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
      ( ( B = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),B2) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_457_boolean__algebra__cancel_Osup2,axiom,
    ! [B: nat,K: nat,B2: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_458_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_o,K: set_o,B2: set_o,A: set_o] :
      ( ( B = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,K),B2) )
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,K),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_459_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_int,K: set_int,B2: set_int,A: set_int] :
      ( ( B = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,K),B2) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,K),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_460_boolean__algebra__cancel_Osup2,axiom,
    ! [B: filter_nat,K: filter_nat,B2: filter_nat,A: filter_nat] :
      ( ( B = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,K),B2) )
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,K),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_461_dual__order_Orefl,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% dual_order.refl
tff(fact_462_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_463_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_464_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_465_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_466_order__refl,axiom,
    ! [Xa: set_int] : aa_set_int_o(ord_less_eq_set_int(Xa),Xa) ).

% order_refl
tff(fact_467_order__refl,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Xa) ).

% order_refl
tff(fact_468_order__refl,axiom,
    ! [Xa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Xa) ).

% order_refl
tff(fact_469_order__refl,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Xa) ).

% order_refl
tff(fact_470_order__refl,axiom,
    ! [Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xa) ).

% order_refl
tff(fact_471_subsetI,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ! [X2: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
         => aa_set3337841544853333262enat_o(member_Extended_enat(X2),B) )
     => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B) ) ).

% subsetI
tff(fact_472_subsetI,axiom,
    ! [A2: set_real,B: set_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => aa_set_real_o(member_real(X2),B) )
     => aa_set_real_o(ord_less_eq_set_real(A2),B) ) ).

% subsetI
tff(fact_473_subsetI,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => aa_set_set_nat_o(member_set_nat(X2),B) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B) ) ).

% subsetI
tff(fact_474_subsetI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_set_nat_o(member_nat(X2),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ).

% subsetI
tff(fact_475_subsetI,axiom,
    ! [A2: set_o,B: set_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A2)
         => aa_set_o_o(member_o((X2)),B) )
     => aa_set_o_o(ord_less_eq_set_o(A2),B) ) ).

% subsetI
tff(fact_476_subsetI,axiom,
    ! [A2: set_int,B: set_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => aa_set_int_o(member_int(X2),B) )
     => aa_set_int_o(ord_less_eq_set_int(A2),B) ) ).

% subsetI
tff(fact_477_subset__antisym,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),A2)
       => ( A2 = B ) ) ) ).

% subset_antisym
tff(fact_478_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_479_le__sup__iff,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),Z)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),Z)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ya),Z) ) ) ).

% le_sup_iff
tff(fact_480_le__sup__iff,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),Z)
    <=> ( aa_set_o_o(ord_less_eq_set_o(Xa),Z)
        & aa_set_o_o(ord_less_eq_set_o(Ya),Z) ) ) ).

% le_sup_iff
tff(fact_481_le__sup__iff,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya),Z)
    <=> ( ord_le2510731241096832064er_nat(Xa,Z)
        & ord_le2510731241096832064er_nat(Ya,Z) ) ) ).

% le_sup_iff
tff(fact_482_le__sup__iff,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),Z)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Z)
        & aa_set_int_o(ord_less_eq_set_int(Ya),Z) ) ) ).

% le_sup_iff
tff(fact_483_le__sup__iff,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ) ).

% le_sup_iff
tff(fact_484_le__sup__iff,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Z) ) ) ).

% le_sup_iff
tff(fact_485_le__sup__iff,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z) ) ) ).

% le_sup_iff
tff(fact_486_sup_Obounded__iff,axiom,
    ! [B2: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_487_sup_Obounded__iff,axiom,
    ! [B2: set_o,C2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)),A)
    <=> ( aa_set_o_o(ord_less_eq_set_o(B2),A)
        & aa_set_o_o(ord_less_eq_set_o(C2),A) ) ) ).

% sup.bounded_iff
tff(fact_488_sup_Obounded__iff,axiom,
    ! [B2: filter_nat,C2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C2),A)
    <=> ( ord_le2510731241096832064er_nat(B2,A)
        & ord_le2510731241096832064er_nat(C2,A) ) ) ).

% sup.bounded_iff
tff(fact_489_sup_Obounded__iff,axiom,
    ! [B2: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B2),A)
        & aa_set_int_o(ord_less_eq_set_int(C2),A) ) ) ).

% sup.bounded_iff
tff(fact_490_sup_Obounded__iff,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B2),C2)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_491_sup_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_492_sup_Obounded__iff,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_493_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_494_subset__empty,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_495_subset__empty,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% subset_empty
tff(fact_496_subset__empty,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_497_subset__empty,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),bot_bo7653980558646680370d_enat)
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% subset_empty
tff(fact_498_subset__empty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_499_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_500_empty__subsetI,axiom,
    ! [A2: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A2) ).

% empty_subsetI
tff(fact_501_empty__subsetI,axiom,
    ! [A2: set_set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(bot_bot_set_set_nat),A2) ).

% empty_subsetI
tff(fact_502_empty__subsetI,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% empty_subsetI
tff(fact_503_empty__subsetI,axiom,
    ! [A2: set_Extended_enat] : aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat),A2) ).

% empty_subsetI
tff(fact_504_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_505_insert__subset,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(insert_Extended_enat(Xa,A2)),B)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B)
        & aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B) ) ) ).

% insert_subset
tff(fact_506_insert__subset,axiom,
    ! [Xa: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(Xa,A2)),B)
    <=> ( aa_set_real_o(member_real(Xa),B)
        & aa_set_real_o(ord_less_eq_set_real(A2),B) ) ) ).

% insert_subset
tff(fact_507_insert__subset,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(Xa,A2)),B)
    <=> ( aa_set_set_nat_o(member_set_nat(Xa),B)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B) ) ) ).

% insert_subset
tff(fact_508_insert__subset,axiom,
    ! [Xa: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(Xa,A2)),B)
    <=> ( aa_set_nat_o(member_nat(Xa),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ) ).

% insert_subset
tff(fact_509_insert__subset,axiom,
    ! [Xa: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o((Xa),A2)),B)
    <=> ( aa_set_o_o(member_o((Xa)),B)
        & aa_set_o_o(ord_less_eq_set_o(A2),B) ) ) ).

% insert_subset
tff(fact_510_insert__subset,axiom,
    ! [Xa: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(Xa,A2)),B)
    <=> ( aa_set_int_o(member_int(Xa),B)
        & aa_set_int_o(ord_less_eq_set_int(A2),B) ) ) ).

% insert_subset
tff(fact_511_Un__subset__iff,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),C)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C) ) ) ).

% Un_subset_iff
tff(fact_512_Un__subset__iff,axiom,
    ! [A2: set_o,B: set_o,C: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),C)
    <=> ( aa_set_o_o(ord_less_eq_set_o(A2),C)
        & aa_set_o_o(ord_less_eq_set_o(B),C) ) ) ).

% Un_subset_iff
tff(fact_513_Un__subset__iff,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),C)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),C)
        & aa_set_int_o(ord_less_eq_set_int(B),C) ) ) ).

% Un_subset_iff
tff(fact_514_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B2,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B2 )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(B2,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_515_singleton__insert__inj__eq,axiom,
    ! [B2: $o,A: $o,A2: set_o] :
      ( ( insert_o((B2),bot_bot_set_o) = insert_o((A),A2) )
    <=> ( ( (A)
        <=> (B2) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B2),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_516_singleton__insert__inj__eq,axiom,
    ! [B2: set_nat,A: set_nat,A2: set_set_nat] :
      ( ( insert_set_nat(B2,bot_bot_set_set_nat) = insert_set_nat(A,A2) )
    <=> ( ( A = B2 )
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(B2,bot_bot_set_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_517_singleton__insert__inj__eq,axiom,
    ! [B2: real,A: real,A2: set_real] :
      ( ( insert_real(B2,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B2 )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B2,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_518_singleton__insert__inj__eq,axiom,
    ! [B2: extended_enat,A: extended_enat,A2: set_Extended_enat] :
      ( ( insert_Extended_enat(B2,bot_bo7653980558646680370d_enat) = insert_Extended_enat(A,A2) )
    <=> ( ( A = B2 )
        & aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(B2,bot_bo7653980558646680370d_enat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_519_singleton__insert__inj__eq,axiom,
    ! [B2: int,A: int,A2: set_int] :
      ( ( insert_int(B2,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B2 )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B2,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_520_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B2: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B2,bot_bot_set_nat) )
    <=> ( ( A = B2 )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(B2,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_521_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B2: $o] :
      ( ( insert_o((A),A2) = insert_o((B2),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B2) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B2),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_522_singleton__insert__inj__eq_H,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_nat] :
      ( ( insert_set_nat(A,A2) = insert_set_nat(B2,bot_bot_set_set_nat) )
    <=> ( ( A = B2 )
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(B2,bot_bot_set_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_523_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B2: real] :
      ( ( insert_real(A,A2) = insert_real(B2,bot_bot_set_real) )
    <=> ( ( A = B2 )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B2,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_524_singleton__insert__inj__eq_H,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B2: extended_enat] :
      ( ( insert_Extended_enat(A,A2) = insert_Extended_enat(B2,bot_bo7653980558646680370d_enat) )
    <=> ( ( A = B2 )
        & aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(B2,bot_bo7653980558646680370d_enat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_525_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B2: int] :
      ( ( insert_int(A,A2) = insert_int(B2,bot_bot_set_int) )
    <=> ( ( A = B2 )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B2,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_526_in__mono,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B) ) ) ).

% in_mono
tff(fact_527_in__mono,axiom,
    ! [A2: set_real,B: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
     => ( aa_set_real_o(member_real(Xa),A2)
       => aa_set_real_o(member_real(Xa),B) ) ) ).

% in_mono
tff(fact_528_in__mono,axiom,
    ! [A2: set_set_nat,B: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => aa_set_set_nat_o(member_set_nat(Xa),B) ) ) ).

% in_mono
tff(fact_529_in__mono,axiom,
    ! [A2: set_nat,B: set_nat,Xa: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => aa_set_nat_o(member_nat(Xa),B) ) ) ).

% in_mono
tff(fact_530_in__mono,axiom,
    ! [A2: set_o,B: set_o,Xa: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => aa_set_o_o(member_o((Xa)),B) ) ) ).

% in_mono
tff(fact_531_in__mono,axiom,
    ! [A2: set_int,B: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(member_int(Xa),A2)
       => aa_set_int_o(member_int(Xa),B) ) ) ).

% in_mono
tff(fact_532_subsetD,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,C2: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
       => aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% subsetD
tff(fact_533_subsetD,axiom,
    ! [A2: set_real,B: set_real,C2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
     => ( aa_set_real_o(member_real(C2),A2)
       => aa_set_real_o(member_real(C2),B) ) ) ).

% subsetD
tff(fact_534_subsetD,axiom,
    ! [A2: set_set_nat,B: set_set_nat,C2: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
     => ( aa_set_set_nat_o(member_set_nat(C2),A2)
       => aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% subsetD
tff(fact_535_subsetD,axiom,
    ! [A2: set_nat,B: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(member_nat(C2),A2)
       => aa_set_nat_o(member_nat(C2),B) ) ) ).

% subsetD
tff(fact_536_subsetD,axiom,
    ! [A2: set_o,B: set_o,C2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( aa_set_o_o(member_o((C2)),A2)
       => aa_set_o_o(member_o((C2)),B) ) ) ).

% subsetD
tff(fact_537_subsetD,axiom,
    ! [A2: set_int,B: set_int,C2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(member_int(C2),A2)
       => aa_set_int_o(member_int(C2),B) ) ) ).

% subsetD
tff(fact_538_equalityE,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A2),B)
         => ~ aa_set_int_o(ord_less_eq_set_int(B),A2) ) ) ).

% equalityE
tff(fact_539_subset__eq,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
    <=> ! [X: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
         => aa_set3337841544853333262enat_o(member_Extended_enat(X),B) ) ) ).

% subset_eq
tff(fact_540_subset__eq,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => aa_set_real_o(member_real(X),B) ) ) ).

% subset_eq
tff(fact_541_subset__eq,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
    <=> ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A2)
         => aa_set_set_nat_o(member_set_nat(X),B) ) ) ).

% subset_eq
tff(fact_542_subset__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_set_nat_o(member_nat(X),B) ) ) ).

% subset_eq
tff(fact_543_subset__eq,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
    <=> ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A2)
         => aa_set_o_o(member_o((X)),B) ) ) ).

% subset_eq
tff(fact_544_subset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_set_int_o(member_int(X),B) ) ) ).

% subset_eq
tff(fact_545_equalityD1,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => aa_set_int_o(ord_less_eq_set_int(A2),B) ) ).

% equalityD1
tff(fact_546_equalityD2,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => aa_set_int_o(ord_less_eq_set_int(B),A2) ) ).

% equalityD2
tff(fact_547_subset__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
    <=> ! [T: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(T),A2)
         => aa_set3337841544853333262enat_o(member_Extended_enat(T),B) ) ) ).

% subset_iff
tff(fact_548_subset__iff,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real(T),A2)
         => aa_set_real_o(member_real(T),B) ) ) ).

% subset_iff
tff(fact_549_subset__iff,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
    <=> ! [T: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T),A2)
         => aa_set_set_nat_o(member_set_nat(T),B) ) ) ).

% subset_iff
tff(fact_550_subset__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A2)
         => aa_set_nat_o(member_nat(T),B) ) ) ).

% subset_iff
tff(fact_551_subset__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
    <=> ! [T: $o] :
          ( aa_set_o_o(member_o((T)),A2)
         => aa_set_o_o(member_o((T)),B) ) ) ).

% subset_iff
tff(fact_552_subset__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A2)
         => aa_set_int_o(member_int(T),B) ) ) ).

% subset_iff
tff(fact_553_subset__refl,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(A2),A2) ).

% subset_refl
tff(fact_554_Collect__mono,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X2: real] :
          ( aa_real_o(P,X2)
         => aa_real_o(Q,X2) )
     => aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q)) ) ).

% Collect_mono
tff(fact_555_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X2: list_nat] :
          ( aa_list_nat_o(P,X2)
         => aa_list_nat_o(Q,X2) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_556_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
         => aa_set_nat_o(Q,X2) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_557_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
         => aa_nat_o(Q,X2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_558_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
         => aa_int_o(Q,X2) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_559_subset__trans,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_eq_set_int(A2),C) ) ) ).

% subset_trans
tff(fact_560_set__eq__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B)
        & aa_set_int_o(ord_less_eq_set_int(B),A2) ) ) ).

% set_eq_subset
tff(fact_561_Collect__mono__iff,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q))
    <=> ! [X: real] :
          ( aa_real_o(P,X)
         => aa_real_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_562_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X: list_nat] :
          ( aa_list_nat_o(P,X)
         => aa_list_nat_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_563_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))
    <=> ! [X: set_nat] :
          ( aa_set_nat_o(P,X)
         => aa_set_nat_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_564_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))
    <=> ! [X: nat] :
          ( aa_nat_o(P,X)
         => aa_nat_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_565_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q))
    <=> ! [X: int] :
          ( aa_int_o(P,X)
         => aa_int_o(Q,X) ) ) ).

% Collect_mono_iff
tff(fact_566_order__antisym__conv,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
     => ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% order_antisym_conv
tff(fact_567_order__antisym__conv,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% order_antisym_conv
tff(fact_568_order__antisym__conv,axiom,
    ! [Ya: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% order_antisym_conv
tff(fact_569_order__antisym__conv,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% order_antisym_conv
tff(fact_570_order__antisym__conv,axiom,
    ! [Ya: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% order_antisym_conv
tff(fact_571_linorder__le__cases,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa) ) ).

% linorder_le_cases
tff(fact_572_linorder__le__cases,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa) ) ).

% linorder_le_cases
tff(fact_573_linorder__le__cases,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa) ) ).

% linorder_le_cases
tff(fact_574_linorder__le__cases,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa) ) ).

% linorder_le_cases
tff(fact_575_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_rat(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_576_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_num(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_577_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_nat(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_578_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_int(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_579_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_rat(F,B2) = C2 )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_580_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_num(F,B2) = C2 )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_581_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_nat(F,B2) = C2 )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_582_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_int(F,B2) = C2 )
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_583_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( aa_nat_rat(F,B2) = C2 )
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_584_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( aa_nat_num(F,B2) = C2 )
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_585_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_586_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_587_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_588_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B2: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_589_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C2: num] :
      ( ( A = aa_num_rat(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_590_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B2: num,C2: num] :
      ( ( A = aa_num_num(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_591_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B2: num,C2: num] :
      ( ( A = aa_num_nat(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_592_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B2: num,C2: num] :
      ( ( A = aa_num_int(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_593_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( ( A = aa_nat_rat(F,B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_594_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B2: nat,C2: nat] :
      ( ( A = aa_nat_num(F,B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_595_linorder__linear,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa) ) ).

% linorder_linear
tff(fact_596_linorder__linear,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa) ) ).

% linorder_linear
tff(fact_597_linorder__linear,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa) ) ).

% linorder_linear
tff(fact_598_linorder__linear,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa) ) ).

% linorder_linear
tff(fact_599_order__eq__refl,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ( Xa = Ya )
     => aa_set_int_o(ord_less_eq_set_int(Xa),Ya) ) ).

% order_eq_refl
tff(fact_600_order__eq__refl,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( Xa = Ya )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ).

% order_eq_refl
tff(fact_601_order__eq__refl,axiom,
    ! [Xa: num,Ya: num] :
      ( ( Xa = Ya )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya) ) ).

% order_eq_refl
tff(fact_602_order__eq__refl,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( Xa = Ya )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya) ) ).

% order_eq_refl
tff(fact_603_order__eq__refl,axiom,
    ! [Xa: int,Ya: int] :
      ( ( Xa = Ya )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya) ) ).

% order_eq_refl
tff(fact_604_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_605_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_606_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_607_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_608_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_609_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_610_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_611_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_612_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B2)),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_613_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B2)),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_614_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_615_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_616_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_617_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B2: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_618_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_619_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_620_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B2: nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_621_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B2: int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X2)),aa_int_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_622_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_623_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B2: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_624_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & aa_set_int_o(ord_less_eq_set_int(B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_625_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_626_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_627_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_628_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_629_antisym,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_630_antisym,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_631_antisym,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_632_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_633_antisym,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_634_dual__order_Otrans,axiom,
    ! [B2: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),B2)
       => aa_set_int_o(ord_less_eq_set_int(C2),A) ) ) ).

% dual_order.trans
tff(fact_635_dual__order_Otrans,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.trans
tff(fact_636_dual__order_Otrans,axiom,
    ! [B2: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( 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),A) ) ) ).

% dual_order.trans
tff(fact_637_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.trans
tff(fact_638_dual__order_Otrans,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( 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),A) ) ) ).

% dual_order.trans
tff(fact_639_dual__order_Oantisym,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(ord_less_eq_set_int(A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_640_dual__order_Oantisym,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_641_dual__order_Oantisym,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_642_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_643_dual__order_Oantisym,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_644_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = B2 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(B2),A)
        & aa_set_int_o(ord_less_eq_set_int(A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_645_dual__order_Oeq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_646_dual__order_Oeq__iff,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_647_dual__order_Oeq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_648_dual__order_Oeq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_649_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B2: rat] :
      ( ! [A3: rat,B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B4)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B4) )
     => ( ! [A3: rat,B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B4),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P,A3),B4) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B2) ) ) ).

% linorder_wlog
tff(fact_650_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B2: num] :
      ( ! [A3: num,B4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B4)
         => aa_num_o(aa_num_fun_num_o(P,A3),B4) )
     => ( ! [A3: num,B4: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B4),A3)
           => aa_num_o(aa_num_fun_num_o(P,A3),B4) )
       => aa_num_o(aa_num_fun_num_o(P,A),B2) ) ) ).

% linorder_wlog
tff(fact_651_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B2: nat] :
      ( ! [A3: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B4)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B4) )
     => ( ! [A3: nat,B4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B4) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B2) ) ) ).

% linorder_wlog
tff(fact_652_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B2: int] :
      ( ! [A3: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B4)
         => aa_int_o(aa_int_fun_int_o(P,A3),B4) )
     => ( ! [A3: int,B4: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B4),A3)
           => aa_int_o(aa_int_fun_int_o(P,A3),B4) )
       => aa_int_o(aa_int_fun_int_o(P,A),B2) ) ) ).

% linorder_wlog
tff(fact_653_order__trans,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_eq_set_int(Ya),Z)
       => aa_set_int_o(ord_less_eq_set_int(Xa),Z) ) ) ).

% order_trans
tff(fact_654_order__trans,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),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,Xa),Z) ) ) ).

% order_trans
tff(fact_655_order__trans,axiom,
    ! [Xa: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),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,Xa),Z) ) ) ).

% order_trans
tff(fact_656_order__trans,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),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,Xa),Z) ) ) ).

% order_trans
tff(fact_657_order__trans,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),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,Xa),Z) ) ) ).

% order_trans
tff(fact_658_order_Otrans,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% order.trans
tff(fact_659_order_Otrans,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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,A),C2) ) ) ).

% order.trans
tff(fact_660_order_Otrans,axiom,
    ! [A: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),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,A),C2) ) ) ).

% order.trans
tff(fact_661_order_Otrans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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,A),C2) ) ) ).

% order.trans
tff(fact_662_order_Otrans,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% order.trans
tff(fact_663_order__antisym,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
       => ( Xa = Ya ) ) ) ).

% order_antisym
tff(fact_664_order__antisym,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
       => ( Xa = Ya ) ) ) ).

% order_antisym
tff(fact_665_order__antisym,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa)
       => ( Xa = Ya ) ) ) ).

% order_antisym
tff(fact_666_order__antisym,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
       => ( Xa = Ya ) ) ) ).

% order_antisym
tff(fact_667_order__antisym,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
       => ( Xa = Ya ) ) ) ).

% order_antisym
tff(fact_668_ord__le__eq__trans,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( ( B2 = C2 )
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_669_ord__le__eq__trans,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( B2 = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_670_ord__le__eq__trans,axiom,
    ! [A: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( B2 = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_671_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( B2 = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_672_ord__le__eq__trans,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( ( B2 = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_673_ord__eq__le__trans,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( ( A = B2 )
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_674_ord__eq__le__trans,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_675_ord__eq__le__trans,axiom,
    ! [A: num,B2: num,C2: num] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_676_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_677_ord__eq__le__trans,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_678_order__class_Oorder__eq__iff,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ( Xa = Ya )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
        & aa_set_int_o(ord_less_eq_set_int(Ya),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_679_order__class_Oorder__eq__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( Xa = Ya )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_680_order__class_Oorder__eq__iff,axiom,
    ! [Xa: num,Ya: num] :
      ( ( Xa = Ya )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_681_order__class_Oorder__eq__iff,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( Xa = Ya )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_682_order__class_Oorder__eq__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( ( Xa = Ya )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_683_le__cases3,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),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),Xa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),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),Xa) )
           => ( ( 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),Xa) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_684_le__cases3,axiom,
    ! [Xa: num,Ya: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),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),Xa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),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),Xa) )
           => ( ( 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),Xa) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_685_le__cases3,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),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),Xa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),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),Xa) )
           => ( ( 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),Xa) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_686_le__cases3,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),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),Xa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),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),Xa) )
           => ( ( 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),Xa) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_687_nle__le,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_688_nle__le,axiom,
    ! [A: num,B2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_689_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_690_nle__le,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_691_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_692_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_693_bot_Oextremum,axiom,
    ! [A: filter_nat] : ord_le2510731241096832064er_nat(bot_bot_filter_nat,A) ).

% bot.extremum
tff(fact_694_bot_Oextremum,axiom,
    ! [A: set_set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(bot_bot_set_set_nat),A) ).

% bot.extremum
tff(fact_695_bot_Oextremum,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% bot.extremum
tff(fact_696_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat),A) ).

% bot.extremum
tff(fact_697_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_698_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_699_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_700_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_701_bot_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
    <=> ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_702_bot_Oextremum__unique,axiom,
    ! [A: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),bot_bot_set_set_nat)
    <=> ( A = bot_bot_set_set_nat ) ) ).

% bot.extremum_unique
tff(fact_703_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_704_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat)
    <=> ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
tff(fact_705_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_706_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_707_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_708_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_709_bot_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
     => ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_710_bot_Oextremum__uniqueI,axiom,
    ! [A: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A),bot_bot_set_set_nat)
     => ( A = bot_bot_set_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_711_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_712_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat)
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
tff(fact_713_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_714_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_715_inf__sup__ord_I4_J,axiom,
    ! [Ya: set_nat,Xa: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ya),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_716_inf__sup__ord_I4_J,axiom,
    ! [Ya: set_o,Xa: set_o] : aa_set_o_o(ord_less_eq_set_o(Ya),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_717_inf__sup__ord_I4_J,axiom,
    ! [Ya: filter_nat,Xa: filter_nat] : ord_le2510731241096832064er_nat(Ya,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_718_inf__sup__ord_I4_J,axiom,
    ! [Ya: set_int,Xa: set_int] : aa_set_int_o(ord_less_eq_set_int(Ya),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_719_inf__sup__ord_I4_J,axiom,
    ! [Ya: rat,Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_720_inf__sup__ord_I4_J,axiom,
    ! [Ya: nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_721_inf__sup__ord_I4_J,axiom,
    ! [Ya: int,Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya)) ).

% inf_sup_ord(4)
tff(fact_722_inf__sup__ord_I3_J,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_723_inf__sup__ord_I3_J,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_o(ord_less_eq_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_724_inf__sup__ord_I3_J,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : ord_le2510731241096832064er_nat(Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_725_inf__sup__ord_I3_J,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_726_inf__sup__ord_I3_J,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_727_inf__sup__ord_I3_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_728_inf__sup__ord_I3_J,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya)) ).

% inf_sup_ord(3)
tff(fact_729_le__supE,axiom,
    ! [A: set_nat,B2: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)),Xa)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),Xa)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),Xa) ) ) ).

% le_supE
tff(fact_730_le__supE,axiom,
    ! [A: set_o,B2: set_o,Xa: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)),Xa)
     => ~ ( aa_set_o_o(ord_less_eq_set_o(A),Xa)
         => ~ aa_set_o_o(ord_less_eq_set_o(B2),Xa) ) ) ).

% le_supE
tff(fact_731_le__supE,axiom,
    ! [A: filter_nat,B2: filter_nat,Xa: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2),Xa)
     => ~ ( ord_le2510731241096832064er_nat(A,Xa)
         => ~ ord_le2510731241096832064er_nat(B2,Xa) ) ) ).

% le_supE
tff(fact_732_le__supE,axiom,
    ! [A: set_int,B2: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)),Xa)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A),Xa)
         => ~ aa_set_int_o(ord_less_eq_set_int(B2),Xa) ) ) ).

% le_supE
tff(fact_733_le__supE,axiom,
    ! [A: rat,B2: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)),Xa)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Xa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),Xa) ) ) ).

% le_supE
tff(fact_734_le__supE,axiom,
    ! [A: nat,B2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)),Xa)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Xa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),Xa) ) ) ).

% le_supE
tff(fact_735_le__supE,axiom,
    ! [A: int,B2: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)),Xa)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),Xa) ) ) ).

% le_supE
tff(fact_736_le__supI,axiom,
    ! [A: set_nat,Xa: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),Xa)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),Xa)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)),Xa) ) ) ).

% le_supI
tff(fact_737_le__supI,axiom,
    ! [A: set_o,Xa: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),Xa)
     => ( aa_set_o_o(ord_less_eq_set_o(B2),Xa)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)),Xa) ) ) ).

% le_supI
tff(fact_738_le__supI,axiom,
    ! [A: filter_nat,Xa: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,Xa)
     => ( ord_le2510731241096832064er_nat(B2,Xa)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2),Xa) ) ) ).

% le_supI
tff(fact_739_le__supI,axiom,
    ! [A: set_int,Xa: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),Xa)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),Xa)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)),Xa) ) ) ).

% le_supI
tff(fact_740_le__supI,axiom,
    ! [A: rat,Xa: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)),Xa) ) ) ).

% le_supI
tff(fact_741_le__supI,axiom,
    ! [A: nat,Xa: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),Xa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)),Xa) ) ) ).

% le_supI
tff(fact_742_le__supI,axiom,
    ! [A: int,Xa: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),Xa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)),Xa) ) ) ).

% le_supI
tff(fact_743_sup__ge1,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) ).

% sup_ge1
tff(fact_744_sup__ge1,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_o(ord_less_eq_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) ).

% sup_ge1
tff(fact_745_sup__ge1,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : ord_le2510731241096832064er_nat(Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) ).

% sup_ge1
tff(fact_746_sup__ge1,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) ).

% sup_ge1
tff(fact_747_sup__ge1,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya)) ).

% sup_ge1
tff(fact_748_sup__ge1,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) ).

% sup_ge1
tff(fact_749_sup__ge1,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya)) ).

% sup_ge1
tff(fact_750_sup__ge2,axiom,
    ! [Ya: set_nat,Xa: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ya),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) ).

% sup_ge2
tff(fact_751_sup__ge2,axiom,
    ! [Ya: set_o,Xa: set_o] : aa_set_o_o(ord_less_eq_set_o(Ya),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) ).

% sup_ge2
tff(fact_752_sup__ge2,axiom,
    ! [Ya: filter_nat,Xa: filter_nat] : ord_le2510731241096832064er_nat(Ya,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) ).

% sup_ge2
tff(fact_753_sup__ge2,axiom,
    ! [Ya: set_int,Xa: set_int] : aa_set_int_o(ord_less_eq_set_int(Ya),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) ).

% sup_ge2
tff(fact_754_sup__ge2,axiom,
    ! [Ya: rat,Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya)) ).

% sup_ge2
tff(fact_755_sup__ge2,axiom,
    ! [Ya: nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) ).

% sup_ge2
tff(fact_756_sup__ge2,axiom,
    ! [Ya: int,Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya)) ).

% sup_ge2
tff(fact_757_le__supI1,axiom,
    ! [Xa: set_nat,A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% le_supI1
tff(fact_758_le__supI1,axiom,
    ! [Xa: set_o,A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(Xa),A)
     => aa_set_o_o(ord_less_eq_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% le_supI1
tff(fact_759_le__supI1,axiom,
    ! [Xa: filter_nat,A: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xa,A)
     => ord_le2510731241096832064er_nat(Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% le_supI1
tff(fact_760_le__supI1,axiom,
    ! [Xa: set_int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),A)
     => aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% le_supI1
tff(fact_761_le__supI1,axiom,
    ! [Xa: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% le_supI1
tff(fact_762_le__supI1,axiom,
    ! [Xa: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ).

% le_supI1
tff(fact_763_le__supI1,axiom,
    ! [Xa: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% le_supI1
tff(fact_764_le__supI2,axiom,
    ! [Xa: set_nat,B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% le_supI2
tff(fact_765_le__supI2,axiom,
    ! [Xa: set_o,B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(Xa),B2)
     => aa_set_o_o(ord_less_eq_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% le_supI2
tff(fact_766_le__supI2,axiom,
    ! [Xa: filter_nat,B2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xa,B2)
     => ord_le2510731241096832064er_nat(Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% le_supI2
tff(fact_767_le__supI2,axiom,
    ! [Xa: set_int,B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),B2)
     => aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% le_supI2
tff(fact_768_le__supI2,axiom,
    ! [Xa: rat,B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% le_supI2
tff(fact_769_le__supI2,axiom,
    ! [Xa: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ).

% le_supI2
tff(fact_770_le__supI2,axiom,
    ! [Xa: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% le_supI2
tff(fact_771_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D),B2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ) ).

% sup.mono
tff(fact_772_sup_Omono,axiom,
    ! [C2: set_o,A: set_o,D: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(C2),A)
     => ( aa_set_o_o(ord_less_eq_set_o(D),B2)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,C2),D)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ) ).

% sup.mono
tff(fact_773_sup_Omono,axiom,
    ! [C2: filter_nat,A: filter_nat,D: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(C2,A)
     => ( ord_le2510731241096832064er_nat(D,B2)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,C2),D),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ) ).

% sup.mono
tff(fact_774_sup_Omono,axiom,
    ! [C2: set_int,A: set_int,D: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(C2),A)
     => ( aa_set_int_o(ord_less_eq_set_int(D),B2)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ) ).

% sup.mono
tff(fact_775_sup_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C2),D)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ) ).

% sup.mono
tff(fact_776_sup_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ) ).

% sup.mono
tff(fact_777_sup_Omono,axiom,
    ! [C2: int,A: int,D: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ) ).

% sup.mono
tff(fact_778_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),D)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D)) ) ) ).

% sup_mono
tff(fact_779_sup__mono,axiom,
    ! [A: set_o,C2: set_o,B2: set_o,D: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),C2)
     => ( aa_set_o_o(ord_less_eq_set_o(B2),D)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,C2),D)) ) ) ).

% sup_mono
tff(fact_780_sup__mono,axiom,
    ! [A: filter_nat,C2: filter_nat,B2: filter_nat,D: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,C2)
     => ( ord_le2510731241096832064er_nat(B2,D)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,C2),D)) ) ) ).

% sup_mono
tff(fact_781_sup__mono,axiom,
    ! [A: set_int,C2: set_int,B2: set_int,D: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),C2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),D)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D)) ) ) ).

% sup_mono
tff(fact_782_sup__mono,axiom,
    ! [A: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C2),D)) ) ) ).

% sup_mono
tff(fact_783_sup__mono,axiom,
    ! [A: nat,C2: nat,B2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D)) ) ) ).

% sup_mono
tff(fact_784_sup__mono,axiom,
    ! [A: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D)) ) ) ).

% sup_mono
tff(fact_785_sup__least,axiom,
    ! [Ya: set_nat,Xa: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ya),Xa)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),Xa)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)),Xa) ) ) ).

% sup_least
tff(fact_786_sup__least,axiom,
    ! [Ya: set_o,Xa: set_o,Z: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(Ya),Xa)
     => ( aa_set_o_o(ord_less_eq_set_o(Z),Xa)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)),Xa) ) ) ).

% sup_least
tff(fact_787_sup__least,axiom,
    ! [Ya: filter_nat,Xa: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(Ya,Xa)
     => ( ord_le2510731241096832064er_nat(Z,Xa)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z),Xa) ) ) ).

% sup_least
tff(fact_788_sup__least,axiom,
    ! [Ya: set_int,Xa: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
     => ( aa_set_int_o(ord_less_eq_set_int(Z),Xa)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)),Xa) ) ) ).

% sup_least
tff(fact_789_sup__least,axiom,
    ! [Ya: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Ya),Z)),Xa) ) ) ).

% sup_least
tff(fact_790_sup__least,axiom,
    ! [Ya: nat,Xa: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)),Xa) ) ) ).

% sup_least
tff(fact_791_sup__least,axiom,
    ! [Ya: int,Xa: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Ya),Z)),Xa) ) ) ).

% sup_least
tff(fact_792_le__iff__sup,axiom,
    ! [Xa: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),Ya)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_793_le__iff__sup,axiom,
    ! [Xa: set_o,Ya: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(Xa),Ya)
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_794_le__iff__sup,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xa,Ya)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_795_le__iff__sup,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_796_le__iff__sup,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_797_le__iff__sup,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_798_le__iff__sup,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya) = Ya ) ) ).

% le_iff_sup
tff(fact_799_sup_OorderE,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) ) ) ).

% sup.orderE
tff(fact_800_sup_OorderE,axiom,
    ! [B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A)
     => ( A = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) ) ) ).

% sup.orderE
tff(fact_801_sup_OorderE,axiom,
    ! [B2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A)
     => ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) ) ) ).

% sup.orderE
tff(fact_802_sup_OorderE,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) ) ) ).

% sup.orderE
tff(fact_803_sup_OorderE,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) ) ) ).

% sup.orderE
tff(fact_804_sup_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) ) ) ).

% sup.orderE
tff(fact_805_sup_OorderE,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) ) ) ).

% sup.orderE
tff(fact_806_sup_OorderI,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A) ) ).

% sup.orderI
tff(fact_807_sup_OorderI,axiom,
    ! [A: set_o,B2: set_o] :
      ( ( A = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) )
     => aa_set_o_o(ord_less_eq_set_o(B2),A) ) ).

% sup.orderI
tff(fact_808_sup_OorderI,axiom,
    ! [A: filter_nat,B2: filter_nat] :
      ( ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) )
     => ord_le2510731241096832064er_nat(B2,A) ) ).

% sup.orderI
tff(fact_809_sup_OorderI,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) )
     => aa_set_int_o(ord_less_eq_set_int(B2),A) ) ).

% sup.orderI
tff(fact_810_sup_OorderI,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% sup.orderI
tff(fact_811_sup_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ).

% sup.orderI
tff(fact_812_sup_OorderI,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% sup.orderI
tff(fact_813_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,Xa: set_nat,Ya: set_nat] :
      ( ! [X2: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y))
     => ( ! [X2: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y))
       => ( ! [X2: set_nat,Y: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z2),X2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y),Z2)),X2) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_814_sup__unique,axiom,
    ! [F: fun_se2621318847353125966_set_o,Xa: set_o,Ya: set_o] :
      ( ! [X2: set_o,Y: set_o] : aa_set_o_o(ord_less_eq_set_o(X2),aa_set_o_set_o(aa_set2146880807996572927_set_o(F,X2),Y))
     => ( ! [X2: set_o,Y: set_o] : aa_set_o_o(ord_less_eq_set_o(Y),aa_set_o_set_o(aa_set2146880807996572927_set_o(F,X2),Y))
       => ( ! [X2: set_o,Y: set_o,Z2: set_o] :
              ( aa_set_o_o(ord_less_eq_set_o(Y),X2)
             => ( aa_set_o_o(ord_less_eq_set_o(Z2),X2)
               => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(F,Y),Z2)),X2) ) )
         => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = aa_set_o_set_o(aa_set2146880807996572927_set_o(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_815_sup__unique,axiom,
    ! [F: fun_fi7557781610047541018er_nat,Xa: filter_nat,Ya: filter_nat] :
      ( ! [X2: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(X2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X2),Y))
     => ( ! [X2: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(Y,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,X2),Y))
       => ( ! [X2: filter_nat,Y: filter_nat,Z2: filter_nat] :
              ( ord_le2510731241096832064er_nat(Y,X2)
             => ( ord_le2510731241096832064er_nat(Z2,X2)
               => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,Y),Z2),X2) ) )
         => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_816_sup__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,Xa: set_int,Ya: set_int] :
      ( ! [X2: set_int,Y: set_int] : aa_set_int_o(ord_less_eq_set_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X2),Y))
     => ( ! [X2: set_int,Y: set_int] : aa_set_int_o(ord_less_eq_set_int(Y),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X2),Y))
       => ( ! [X2: set_int,Y: set_int,Z2: set_int] :
              ( aa_set_int_o(ord_less_eq_set_int(Y),X2)
             => ( aa_set_int_o(ord_less_eq_set_int(Z2),X2)
               => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y),Z2)),X2) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_817_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,Xa: rat,Ya: rat] :
      ( ! [X2: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y))
     => ( ! [X2: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y))
       => ( ! [X2: rat,Y: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X2)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y),Z2)),X2) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_818_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,Xa: nat,Ya: nat] :
      ( ! [X2: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y))
     => ( ! [X2: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y))
       => ( ! [X2: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y),Z2)),X2) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_819_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,Xa: int,Ya: int] :
      ( ! [X2: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,X2),Y))
     => ( ! [X2: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(F,X2),Y))
       => ( ! [X2: int,Y: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y),Z2)),X2) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya) = aa_int_int(aa_int_fun_int_int(F,Xa),Ya) ) ) ) ) ).

% sup_unique
tff(fact_820_sup_Oabsorb1,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_821_sup_Oabsorb1,axiom,
    ! [B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_822_sup_Oabsorb1,axiom,
    ! [B2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_823_sup_Oabsorb1,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_824_sup_Oabsorb1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_825_sup_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_826_sup_Oabsorb1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = A ) ) ).

% sup.absorb1
tff(fact_827_sup_Oabsorb2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_828_sup_Oabsorb2,axiom,
    ! [A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),B2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_829_sup_Oabsorb2,axiom,
    ! [A: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B2)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_830_sup_Oabsorb2,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_831_sup_Oabsorb2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_832_sup_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_833_sup_Oabsorb2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_834_sup__absorb1,axiom,
    ! [Ya: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ya),Xa)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_835_sup__absorb1,axiom,
    ! [Ya: set_o,Xa: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(Ya),Xa)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_836_sup__absorb1,axiom,
    ! [Ya: filter_nat,Xa: filter_nat] :
      ( ord_le2510731241096832064er_nat(Ya,Xa)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_837_sup__absorb1,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_838_sup__absorb1,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_839_sup__absorb1,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_840_sup__absorb1,axiom,
    ! [Ya: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya) = Xa ) ) ).

% sup_absorb1
tff(fact_841_sup__absorb2,axiom,
    ! [Xa: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),Ya)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_842_sup__absorb2,axiom,
    ! [Xa: set_o,Ya: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(Xa),Ya)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_843_sup__absorb2,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xa,Ya)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_844_sup__absorb2,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_845_sup__absorb2,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_846_sup__absorb2,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_847_sup__absorb2,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya) = Ya ) ) ).

% sup_absorb2
tff(fact_848_sup_OboundedE,axiom,
    ! [B2: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% sup.boundedE
tff(fact_849_sup_OboundedE,axiom,
    ! [B2: set_o,C2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)),A)
     => ~ ( aa_set_o_o(ord_less_eq_set_o(B2),A)
         => ~ aa_set_o_o(ord_less_eq_set_o(C2),A) ) ) ).

% sup.boundedE
tff(fact_850_sup_OboundedE,axiom,
    ! [B2: filter_nat,C2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C2),A)
     => ~ ( ord_le2510731241096832064er_nat(B2,A)
         => ~ ord_le2510731241096832064er_nat(C2,A) ) ) ).

% sup.boundedE
tff(fact_851_sup_OboundedE,axiom,
    ! [B2: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)),A)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(B2),A)
         => ~ aa_set_int_o(ord_less_eq_set_int(C2),A) ) ) ).

% sup.boundedE
tff(fact_852_sup_OboundedE,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B2),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% sup.boundedE
tff(fact_853_sup_OboundedE,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% sup.boundedE
tff(fact_854_sup_OboundedE,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% sup.boundedE
tff(fact_855_sup_OboundedI,axiom,
    ! [B2: set_nat,A: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A) ) ) ).

% sup.boundedI
tff(fact_856_sup_OboundedI,axiom,
    ! [B2: set_o,A: set_o,C2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A)
     => ( aa_set_o_o(ord_less_eq_set_o(C2),A)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)),A) ) ) ).

% sup.boundedI
tff(fact_857_sup_OboundedI,axiom,
    ! [B2: filter_nat,A: filter_nat,C2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A)
     => ( ord_le2510731241096832064er_nat(C2,A)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C2),A) ) ) ).

% sup.boundedI
tff(fact_858_sup_OboundedI,axiom,
    ! [B2: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),A)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)),A) ) ) ).

% sup.boundedI
tff(fact_859_sup_OboundedI,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B2),C2)),A) ) ) ).

% sup.boundedI
tff(fact_860_sup_OboundedI,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C2)),A) ) ) ).

% sup.boundedI
tff(fact_861_sup_OboundedI,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C2)),A) ) ) ).

% sup.boundedI
tff(fact_862_sup_Oorder__iff,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) ) ) ).

% sup.order_iff
tff(fact_863_sup_Oorder__iff,axiom,
    ! [B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A)
    <=> ( A = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) ) ) ).

% sup.order_iff
tff(fact_864_sup_Oorder__iff,axiom,
    ! [B2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A)
    <=> ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) ) ) ).

% sup.order_iff
tff(fact_865_sup_Oorder__iff,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) ) ) ).

% sup.order_iff
tff(fact_866_sup_Oorder__iff,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) ) ) ).

% sup.order_iff
tff(fact_867_sup_Oorder__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) ) ) ).

% sup.order_iff
tff(fact_868_sup_Oorder__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) ) ) ).

% sup.order_iff
tff(fact_869_sup_Ocobounded1,axiom,
    ! [A: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ).

% sup.cobounded1
tff(fact_870_sup_Ocobounded1,axiom,
    ! [A: set_o,B2: set_o] : aa_set_o_o(ord_less_eq_set_o(A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ).

% sup.cobounded1
tff(fact_871_sup_Ocobounded1,axiom,
    ! [A: filter_nat,B2: filter_nat] : ord_le2510731241096832064er_nat(A,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ).

% sup.cobounded1
tff(fact_872_sup_Ocobounded1,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_o(ord_less_eq_set_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ).

% sup.cobounded1
tff(fact_873_sup_Ocobounded1,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ).

% sup.cobounded1
tff(fact_874_sup_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ).

% sup.cobounded1
tff(fact_875_sup_Ocobounded1,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ).

% sup.cobounded1
tff(fact_876_sup_Ocobounded2,axiom,
    ! [B2: set_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ).

% sup.cobounded2
tff(fact_877_sup_Ocobounded2,axiom,
    ! [B2: set_o,A: set_o] : aa_set_o_o(ord_less_eq_set_o(B2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ).

% sup.cobounded2
tff(fact_878_sup_Ocobounded2,axiom,
    ! [B2: filter_nat,A: filter_nat] : ord_le2510731241096832064er_nat(B2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ).

% sup.cobounded2
tff(fact_879_sup_Ocobounded2,axiom,
    ! [B2: set_int,A: set_int] : aa_set_int_o(ord_less_eq_set_int(B2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ).

% sup.cobounded2
tff(fact_880_sup_Ocobounded2,axiom,
    ! [B2: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ).

% sup.cobounded2
tff(fact_881_sup_Ocobounded2,axiom,
    ! [B2: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ).

% sup.cobounded2
tff(fact_882_sup_Ocobounded2,axiom,
    ! [B2: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ).

% sup.cobounded2
tff(fact_883_sup_Oabsorb__iff1,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_884_sup_Oabsorb__iff1,axiom,
    ! [B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A)
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_885_sup_Oabsorb__iff1,axiom,
    ! [B2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_886_sup_Oabsorb__iff1,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_887_sup_Oabsorb__iff1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_888_sup_Oabsorb__iff1,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_889_sup_Oabsorb__iff1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = A ) ) ).

% sup.absorb_iff1
tff(fact_890_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_891_sup_Oabsorb__iff2,axiom,
    ! [A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),B2)
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_892_sup_Oabsorb__iff2,axiom,
    ! [A: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B2)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_893_sup_Oabsorb__iff2,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_894_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_895_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_896_sup_Oabsorb__iff2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_897_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% sup.coboundedI1
tff(fact_898_sup_OcoboundedI1,axiom,
    ! [C2: set_o,A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(C2),A)
     => aa_set_o_o(ord_less_eq_set_o(C2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% sup.coboundedI1
tff(fact_899_sup_OcoboundedI1,axiom,
    ! [C2: filter_nat,A: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(C2,A)
     => ord_le2510731241096832064er_nat(C2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% sup.coboundedI1
tff(fact_900_sup_OcoboundedI1,axiom,
    ! [C2: set_int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(C2),A)
     => aa_set_int_o(ord_less_eq_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% sup.coboundedI1
tff(fact_901_sup_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% sup.coboundedI1
tff(fact_902_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ).

% sup.coboundedI1
tff(fact_903_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% sup.coboundedI1
tff(fact_904_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% sup.coboundedI2
tff(fact_905_sup_OcoboundedI2,axiom,
    ! [C2: set_o,B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(C2),B2)
     => aa_set_o_o(ord_less_eq_set_o(C2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% sup.coboundedI2
tff(fact_906_sup_OcoboundedI2,axiom,
    ! [C2: filter_nat,B2: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(C2,B2)
     => ord_le2510731241096832064er_nat(C2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% sup.coboundedI2
tff(fact_907_sup_OcoboundedI2,axiom,
    ! [C2: set_int,B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(C2),B2)
     => aa_set_int_o(ord_less_eq_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% sup.coboundedI2
tff(fact_908_sup_OcoboundedI2,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% sup.coboundedI2
tff(fact_909_sup_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A: 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(sup_sup_nat,A),B2)) ) ).

% sup.coboundedI2
tff(fact_910_sup_OcoboundedI2,axiom,
    ! [C2: int,B2: int,A: 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(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% sup.coboundedI2
tff(fact_911_insert__mono,axiom,
    ! [C: set_nat,D2: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),D2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(A,C)),insert_nat(A,D2)) ) ).

% insert_mono
tff(fact_912_insert__mono,axiom,
    ! [C: set_o,D2: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C),D2)
     => aa_set_o_o(ord_less_eq_set_o(insert_o((A),C)),insert_o((A),D2)) ) ).

% insert_mono
tff(fact_913_insert__mono,axiom,
    ! [C: set_set_nat,D2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(C),D2)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(A,C)),insert_set_nat(A,D2)) ) ).

% insert_mono
tff(fact_914_insert__mono,axiom,
    ! [C: set_real,D2: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C),D2)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(A,C)),insert_real(A,D2)) ) ).

% insert_mono
tff(fact_915_insert__mono,axiom,
    ! [C: set_Extended_enat,D2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(C),D2)
     => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(insert_Extended_enat(A,C)),insert_Extended_enat(A,D2)) ) ).

% insert_mono
tff(fact_916_insert__mono,axiom,
    ! [C: set_int,D2: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C),D2)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A,C)),insert_int(A,D2)) ) ).

% insert_mono
tff(fact_917_subset__insert,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(Xa,B))
      <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B) ) ) ).

% subset_insert
tff(fact_918_subset__insert,axiom,
    ! [Xa: real,A2: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B))
      <=> aa_set_real_o(ord_less_eq_set_real(A2),B) ) ) ).

% subset_insert
tff(fact_919_subset__insert,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(Xa,B))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B) ) ) ).

% subset_insert
tff(fact_920_subset__insert,axiom,
    ! [Xa: nat,A2: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(Xa,B))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ) ).

% subset_insert
tff(fact_921_subset__insert,axiom,
    ! [Xa: $o,A2: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),B))
      <=> aa_set_o_o(ord_less_eq_set_o(A2),B) ) ) ).

% subset_insert
tff(fact_922_subset__insert,axiom,
    ! [Xa: int,A2: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B))
      <=> aa_set_int_o(ord_less_eq_set_int(A2),B) ) ) ).

% subset_insert
tff(fact_923_subset__insertI,axiom,
    ! [B: set_nat,A: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),insert_nat(A,B)) ).

% subset_insertI
tff(fact_924_subset__insertI,axiom,
    ! [B: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B),insert_o((A),B)) ).

% subset_insertI
tff(fact_925_subset__insertI,axiom,
    ! [B: set_set_nat,A: set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(B),insert_set_nat(A,B)) ).

% subset_insertI
tff(fact_926_subset__insertI,axiom,
    ! [B: set_real,A: real] : aa_set_real_o(ord_less_eq_set_real(B),insert_real(A,B)) ).

% subset_insertI
tff(fact_927_subset__insertI,axiom,
    ! [B: set_Extended_enat,A: extended_enat] : aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B),insert_Extended_enat(A,B)) ).

% subset_insertI
tff(fact_928_subset__insertI,axiom,
    ! [B: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B),insert_int(A,B)) ).

% subset_insertI
tff(fact_929_subset__insertI2,axiom,
    ! [A2: set_nat,B: set_nat,B2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(B2,B)) ) ).

% subset_insertI2
tff(fact_930_subset__insertI2,axiom,
    ! [A2: set_o,B: set_o,B2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B2),B)) ) ).

% subset_insertI2
tff(fact_931_subset__insertI2,axiom,
    ! [A2: set_set_nat,B: set_set_nat,B2: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(B2,B)) ) ).

% subset_insertI2
tff(fact_932_subset__insertI2,axiom,
    ! [A2: set_real,B: set_real,B2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B2,B)) ) ).

% subset_insertI2
tff(fact_933_subset__insertI2,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(B2,B)) ) ).

% subset_insertI2
tff(fact_934_subset__insertI2,axiom,
    ! [A2: set_int,B: set_int,B2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B2,B)) ) ).

% subset_insertI2
tff(fact_935_Un__mono,axiom,
    ! [A2: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)) ) ) ).

% Un_mono
tff(fact_936_Un__mono,axiom,
    ! [A2: set_o,C: set_o,B: set_o,D2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),C)
     => ( aa_set_o_o(ord_less_eq_set_o(B),D2)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,C),D2)) ) ) ).

% Un_mono
tff(fact_937_Un__mono,axiom,
    ! [A2: set_int,C: set_int,B: set_int,D2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),C)
     => ( aa_set_int_o(ord_less_eq_set_int(B),D2)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),D2)) ) ) ).

% Un_mono
tff(fact_938_Un__least,axiom,
    ! [A2: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),C) ) ) ).

% Un_least
tff(fact_939_Un__least,axiom,
    ! [A2: set_o,C: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),C)
     => ( aa_set_o_o(ord_less_eq_set_o(B),C)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),C) ) ) ).

% Un_least
tff(fact_940_Un__least,axiom,
    ! [A2: set_int,C: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),C)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),C) ) ) ).

% Un_least
tff(fact_941_Un__upper1,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) ).

% Un_upper1
tff(fact_942_Un__upper1,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_o(ord_less_eq_set_o(A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) ).

% Un_upper1
tff(fact_943_Un__upper1,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_o(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) ).

% Un_upper1
tff(fact_944_Un__upper2,axiom,
    ! [B: set_nat,A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) ).

% Un_upper2
tff(fact_945_Un__upper2,axiom,
    ! [B: set_o,A2: set_o] : aa_set_o_o(ord_less_eq_set_o(B),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) ).

% Un_upper2
tff(fact_946_Un__upper2,axiom,
    ! [B: set_int,A2: set_int] : aa_set_int_o(ord_less_eq_set_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) ).

% Un_upper2
tff(fact_947_Un__absorb1,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = B ) ) ).

% Un_absorb1
tff(fact_948_Un__absorb1,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = B ) ) ).

% Un_absorb1
tff(fact_949_Un__absorb1,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = B ) ) ).

% Un_absorb1
tff(fact_950_Un__absorb2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = A2 ) ) ).

% Un_absorb2
tff(fact_951_Un__absorb2,axiom,
    ! [B: set_o,A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = A2 ) ) ).

% Un_absorb2
tff(fact_952_Un__absorb2,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = A2 ) ) ).

% Un_absorb2
tff(fact_953_subset__UnE,axiom,
    ! [C: set_nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))
     => ~ ! [A4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),A2)
           => ! [B5: set_nat] :
                ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B5),B)
               => ( C != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5) ) ) ) ) ).

% subset_UnE
tff(fact_954_subset__UnE,axiom,
    ! [C: set_o,A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(C),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
     => ~ ! [A4: set_o] :
            ( aa_set_o_o(ord_less_eq_set_o(A4),A2)
           => ! [B5: set_o] :
                ( aa_set_o_o(ord_less_eq_set_o(B5),B)
               => ( C != aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B5) ) ) ) ) ).

% subset_UnE
tff(fact_955_subset__UnE,axiom,
    ! [C: set_int,A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))
     => ~ ! [A4: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(A4),A2)
           => ! [B5: set_int] :
                ( aa_set_int_o(ord_less_eq_set_int(B5),B)
               => ( C != aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B5) ) ) ) ) ).

% subset_UnE
tff(fact_956_subset__Un__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = B ) ) ).

% subset_Un_eq
tff(fact_957_subset__Un__eq,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
    <=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = B ) ) ).

% subset_Un_eq
tff(fact_958_subset__Un__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = B ) ) ).

% subset_Un_eq
tff(fact_959_subset__singletonD,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_960_subset__singletonD,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),bot_bot_set_o))
     => ( ( A2 = bot_bot_set_o )
        | ( A2 = insert_o((Xa),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_961_subset__singletonD,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(Xa,bot_bot_set_set_nat))
     => ( ( A2 = bot_bot_set_set_nat )
        | ( A2 = insert_set_nat(Xa,bot_bot_set_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_962_subset__singletonD,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,bot_bot_set_real))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(Xa,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_963_subset__singletonD,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))
     => ( ( A2 = bot_bo7653980558646680370d_enat )
        | ( A2 = insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat) ) ) ) ).

% subset_singletonD
tff(fact_964_subset__singletonD,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(Xa,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_965_subset__singleton__iff,axiom,
    ! [X3: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X3 = bot_bot_set_nat )
        | ( X3 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_966_subset__singleton__iff,axiom,
    ! [X3: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X3),insert_o((A),bot_bot_set_o))
    <=> ( ( X3 = bot_bot_set_o )
        | ( X3 = insert_o((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_967_subset__singleton__iff,axiom,
    ! [X3: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(X3),insert_set_nat(A,bot_bot_set_set_nat))
    <=> ( ( X3 = bot_bot_set_set_nat )
        | ( X3 = insert_set_nat(A,bot_bot_set_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_968_subset__singleton__iff,axiom,
    ! [X3: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X3),insert_real(A,bot_bot_set_real))
    <=> ( ( X3 = bot_bot_set_real )
        | ( X3 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_969_subset__singleton__iff,axiom,
    ! [X3: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(X3),insert_Extended_enat(A,bot_bo7653980558646680370d_enat))
    <=> ( ( X3 = bot_bo7653980558646680370d_enat )
        | ( X3 = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ) ) ) ).

% subset_singleton_iff
tff(fact_970_subset__singleton__iff,axiom,
    ! [X3: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X3),insert_int(A,bot_bot_set_int))
    <=> ( ( X3 = bot_bot_set_int )
        | ( X3 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_971_is__singleton__the__elem,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ( A2 = insert_nat(the_elem_nat(A2),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_972_is__singleton__the__elem,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
    <=> ( A2 = insert_int(the_elem_int(A2),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_973_is__singleton__the__elem,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ( A2 = insert_o(aa_set_o_o(the_elem_o,A2),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_974_is__singleton__the__elem,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(is_singleton_set_nat,A2)
    <=> ( A2 = insert_set_nat(the_elem_set_nat(A2),bot_bot_set_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_975_is__singleton__the__elem,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_singleton_real,A2)
    <=> ( A2 = insert_real(the_elem_real(A2),bot_bot_set_real) ) ) ).

% is_singleton_the_elem
tff(fact_976_is__singleton__the__elem,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(is_sin1871519699599484762d_enat,A2)
    <=> ( A2 = insert_Extended_enat(the_el319773668273709403d_enat(A2),bot_bo7653980558646680370d_enat) ) ) ).

% is_singleton_the_elem
tff(fact_977_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X2: nat,Y: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => ( aa_set_nat_o(member_nat(Y),A2)
             => ( X2 = Y ) ) )
       => aa_set_nat_o(is_singleton_nat,A2) ) ) ).

% is_singletonI'
tff(fact_978_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X2: int,Y: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => ( aa_set_int_o(member_int(Y),A2)
             => ( X2 = Y ) ) )
       => aa_set_int_o(is_singleton_int,A2) ) ) ).

% is_singletonI'
tff(fact_979_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X2: $o,Y: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
           => ( aa_set_o_o(member_o((Y)),A2)
             => ( (X2)
              <=> (Y) ) ) )
       => aa_set_o_o(is_singleton_o,A2) ) ) ).

% is_singletonI'
tff(fact_980_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X2: set_nat,Y: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
           => ( aa_set_set_nat_o(member_set_nat(Y),A2)
             => ( X2 = Y ) ) )
       => aa_set_set_nat_o(is_singleton_set_nat,A2) ) ) ).

% is_singletonI'
tff(fact_981_is__singletonI_H,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X2: real,Y: real] :
            ( aa_set_real_o(member_real(X2),A2)
           => ( aa_set_real_o(member_real(Y),A2)
             => ( X2 = Y ) ) )
       => aa_set_real_o(is_singleton_real,A2) ) ) ).

% is_singletonI'
tff(fact_982_is__singletonI_H,axiom,
    ! [A2: set_Extended_enat] :
      ( ( A2 != bot_bo7653980558646680370d_enat )
     => ( ! [X2: extended_enat,Y: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
           => ( aa_set3337841544853333262enat_o(member_Extended_enat(Y),A2)
             => ( X2 = Y ) ) )
       => aa_set3337841544853333262enat_o(is_sin1871519699599484762d_enat,A2) ) ) ).

% is_singletonI'
tff(fact_983_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_984_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_985_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_o,K: set_o,A: set_o,B2: set_o] :
      ( ( A2 = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,K),A) )
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,K),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_986_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_int,K: set_int,A: set_int,B2: set_int] :
      ( ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,K),A) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,K),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_987_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: filter_nat,K: filter_nat,A: filter_nat,B2: filter_nat] :
      ( ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,K),A) )
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,K),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_988_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xa)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xa)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_989_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
     => aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq2
tff(fact_990_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),D)
     => aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).

% valid_eq1
tff(fact_991_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
    <=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq
tff(fact_992_bot__empty__eq,axiom,
    ! [X4: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X4)
    <=> aa_set_nat_o(member_nat(X4),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_993_bot__empty__eq,axiom,
    ! [X4: int] :
      ( aa_int_o(bot_bot_fun_int_o,X4)
    <=> aa_set_int_o(member_int(X4),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_994_bot__empty__eq,axiom,
    ! [X4: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X4))
    <=> aa_set_o_o(member_o((X4)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_995_bot__empty__eq,axiom,
    ! [X4: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X4)
    <=> aa_set_set_nat_o(member_set_nat(X4),bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_996_bot__empty__eq,axiom,
    ! [X4: real] :
      ( aa_real_o(bot_bot_fun_real_o,X4)
    <=> aa_set_real_o(member_real(X4),bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_997_bot__empty__eq,axiom,
    ! [X4: extended_enat] :
      ( aa_Extended_enat_o(bot_bo3575530754339543945enat_o,X4)
    <=> aa_set3337841544853333262enat_o(member_Extended_enat(X4),bot_bo7653980558646680370d_enat) ) ).

% bot_empty_eq
tff(fact_998_Collect__empty__eq__bot,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ( P = bot_bo8828579575001852285_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_999_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1000_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1001_Collect__empty__eq__bot,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ( P = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1002_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo3967814874310145751_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1003_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bot_fun_real_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1004_Collect__empty__eq__bot,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ( collec4429806609662206161d_enat(P) = bot_bo7653980558646680370d_enat )
    <=> ( P = bot_bo3575530754339543945enat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1005_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_1006_insert__subsetI,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,X3: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(X3),A2)
       => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(insert_Extended_enat(Xa,X3)),A2) ) ) ).

% insert_subsetI
tff(fact_1007_insert__subsetI,axiom,
    ! [Xa: real,A2: set_real,X3: set_real] :
      ( aa_set_real_o(member_real(Xa),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(X3),A2)
       => aa_set_real_o(ord_less_eq_set_real(insert_real(Xa,X3)),A2) ) ) ).

% insert_subsetI
tff(fact_1008_insert__subsetI,axiom,
    ! [Xa: set_nat,A2: set_set_nat,X3: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(X3),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(Xa,X3)),A2) ) ) ).

% insert_subsetI
tff(fact_1009_insert__subsetI,axiom,
    ! [Xa: nat,A2: set_nat,X3: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(Xa,X3)),A2) ) ) ).

% insert_subsetI
tff(fact_1010_insert__subsetI,axiom,
    ! [Xa: $o,A2: set_o,X3: set_o] :
      ( aa_set_o_o(member_o((Xa)),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(X3),A2)
       => aa_set_o_o(ord_less_eq_set_o(insert_o((Xa),X3)),A2) ) ) ).

% insert_subsetI
tff(fact_1011_insert__subsetI,axiom,
    ! [Xa: int,A2: set_int,X3: set_int] :
      ( aa_set_int_o(member_int(Xa),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(X3),A2)
       => aa_set_int_o(ord_less_eq_set_int(insert_int(Xa,X3)),A2) ) ) ).

% insert_subsetI
tff(fact_1012_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X2: nat] : ~ aa_set_nat_o(member_nat(X2),A2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1013_subset__emptyI,axiom,
    ! [A2: set_o] :
      ( ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),A2)
     => aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_1014_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X2),A2)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_1015_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X2: real] : ~ aa_set_real_o(member_real(X2),A2)
     => aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real) ) ).

% subset_emptyI
tff(fact_1016_subset__emptyI,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [X2: extended_enat] : ~ aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
     => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),bot_bo7653980558646680370d_enat) ) ).

% subset_emptyI
tff(fact_1017_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X2: int] : ~ aa_set_int_o(member_int(X2),A2)
     => aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1018_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xa)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xa) ) ) ).

% both_member_options_def
tff(fact_1019_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_1020_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_1021_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_1022_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
       => aa_nat_o(vEBT_vebt_member(Ta),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_1023_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xa)
      <=> aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_1024_finite__Un,axiom,
    ! [F2: set_complex,G: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(F2,G))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,F2)
        & aa_set_complex_o(finite3207457112153483333omplex,G) ) ) ).

% finite_Un
tff(fact_1025_finite__Un,axiom,
    ! [F2: set_Extended_enat,G: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(F2,G))
    <=> ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
        & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,G) ) ) ).

% finite_Un
tff(fact_1026_finite__Un,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F2),G))
    <=> ( aa_set_nat_o(finite_finite_nat,F2)
        & aa_set_nat_o(finite_finite_nat,G) ) ) ).

% finite_Un
tff(fact_1027_finite__Un,axiom,
    ! [F2: set_o,G: set_o] :
      ( aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,F2),G))
    <=> ( aa_set_o_o(finite_finite_o,F2)
        & aa_set_o_o(finite_finite_o,G) ) ) ).

% finite_Un
tff(fact_1028_finite__Un,axiom,
    ! [F2: set_int,G: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F2),G))
    <=> ( aa_set_int_o(finite_finite_int,F2)
        & aa_set_int_o(finite_finite_int,G) ) ) ).

% finite_Un
tff(fact_1029_finite__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(finite_finite_o,insert_o((A),A2))
    <=> aa_set_o_o(finite_finite_o,A2) ) ).

% finite_insert
tff(fact_1030_finite__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,insert_set_nat(A,A2))
    <=> aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ).

% finite_insert
tff(fact_1031_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(finite_finite_real,insert_real(A,A2))
    <=> aa_set_real_o(finite_finite_real,A2) ) ).

% finite_insert
tff(fact_1032_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,insert_nat(A,A2))
    <=> aa_set_nat_o(finite_finite_nat,A2) ) ).

% finite_insert
tff(fact_1033_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,insert_int(A,A2))
    <=> aa_set_int_o(finite_finite_int,A2) ) ).

% finite_insert
tff(fact_1034_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A2))
    <=> aa_set_complex_o(finite3207457112153483333omplex,A2) ) ).

% finite_insert
tff(fact_1035_finite__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,insert_Extended_enat(A,A2))
    <=> aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ).

% finite_insert
tff(fact_1036_le__zero__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1037_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xa) ) ).

% min_Null_member
tff(fact_1038_finite__subset__induct_H,axiom,
    ! [F2: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F2),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( aa_set_complex_o(member_complex(A3),A2)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A2)
                   => ( ~ aa_set_complex_o(member_complex(A3),F3)
                     => ( aa_set_complex_o(P,F3)
                       => aa_set_complex_o(P,insert_complex(A3,F3)) ) ) ) ) )
           => aa_set_complex_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1039_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat(A3),A2)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A2)
                   => ( ~ aa_set_nat_o(member_nat(A3),F3)
                     => ( aa_set_nat_o(P,F3)
                       => aa_set_nat_o(P,insert_nat(A3,F3)) ) ) ) ) )
           => aa_set_nat_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1040_finite__subset__induct_H,axiom,
    ! [F2: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(ord_less_eq_set_o(F2),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A2)
                 => ( aa_set_o_o(ord_less_eq_set_o(F3),A2)
                   => ( ~ aa_set_o_o(member_o((A3)),F3)
                     => ( aa_set_o_o(P,F3)
                       => aa_set_o_o(P,insert_o((A3),F3)) ) ) ) ) )
           => aa_set_o_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1041_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F2),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F3: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
               => ( aa_set_set_nat_o(member_set_nat(A3),A2)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A3),F3)
                     => ( aa_set_set_nat_o(P,F3)
                       => aa_set_set_nat_o(P,insert_set_nat(A3,F3)) ) ) ) ) )
           => aa_set_set_nat_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1042_finite__subset__induct_H,axiom,
    ! [F2: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( aa_set_real_o(ord_less_eq_set_real(F2),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( aa_set_real_o(member_real(A3),A2)
                 => ( aa_set_real_o(ord_less_eq_set_real(F3),A2)
                   => ( ~ aa_set_real_o(member_real(A3),F3)
                     => ( aa_set_real_o(P,F3)
                       => aa_set_real_o(P,insert_real(A3,F3)) ) ) ) ) )
           => aa_set_real_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1043_finite__subset__induct_H,axiom,
    ! [F2: set_Extended_enat,A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(F2),A2)
       => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
         => ( ! [A3: extended_enat,F3: set_Extended_enat] :
                ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
               => ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
                 => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(F3),A2)
                   => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A3),F3)
                     => ( aa_set3337841544853333262enat_o(P,F3)
                       => aa_set3337841544853333262enat_o(P,insert_Extended_enat(A3,F3)) ) ) ) ) )
           => aa_set3337841544853333262enat_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1044_finite__subset__induct_H,axiom,
    ! [F2: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(ord_less_eq_set_int(F2),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( aa_set_int_o(member_int(A3),A2)
                 => ( aa_set_int_o(ord_less_eq_set_int(F3),A2)
                   => ( ~ aa_set_int_o(member_int(A3),F3)
                     => ( aa_set_int_o(P,F3)
                       => aa_set_int_o(P,insert_int(A3,F3)) ) ) ) ) )
           => aa_set_int_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1045_finite__subset__induct,axiom,
    ! [F2: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F2),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( aa_set_complex_o(member_complex(A3),A2)
                 => ( ~ aa_set_complex_o(member_complex(A3),F3)
                   => ( aa_set_complex_o(P,F3)
                     => aa_set_complex_o(P,insert_complex(A3,F3)) ) ) ) )
           => aa_set_complex_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1046_finite__subset__induct,axiom,
    ! [F2: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat(A3),A2)
                 => ( ~ aa_set_nat_o(member_nat(A3),F3)
                   => ( aa_set_nat_o(P,F3)
                     => aa_set_nat_o(P,insert_nat(A3,F3)) ) ) ) )
           => aa_set_nat_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1047_finite__subset__induct,axiom,
    ! [F2: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(ord_less_eq_set_o(F2),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A2)
                 => ( ~ aa_set_o_o(member_o((A3)),F3)
                   => ( aa_set_o_o(P,F3)
                     => aa_set_o_o(P,insert_o((A3),F3)) ) ) ) )
           => aa_set_o_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1048_finite__subset__induct,axiom,
    ! [F2: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F2),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F3: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
               => ( aa_set_set_nat_o(member_set_nat(A3),A2)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A3),F3)
                   => ( aa_set_set_nat_o(P,F3)
                     => aa_set_set_nat_o(P,insert_set_nat(A3,F3)) ) ) ) )
           => aa_set_set_nat_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1049_finite__subset__induct,axiom,
    ! [F2: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( aa_set_real_o(ord_less_eq_set_real(F2),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( aa_set_real_o(member_real(A3),A2)
                 => ( ~ aa_set_real_o(member_real(A3),F3)
                   => ( aa_set_real_o(P,F3)
                     => aa_set_real_o(P,insert_real(A3,F3)) ) ) ) )
           => aa_set_real_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1050_finite__subset__induct,axiom,
    ! [F2: set_Extended_enat,A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(F2),A2)
       => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
         => ( ! [A3: extended_enat,F3: set_Extended_enat] :
                ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
               => ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
                 => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A3),F3)
                   => ( aa_set3337841544853333262enat_o(P,F3)
                     => aa_set3337841544853333262enat_o(P,insert_Extended_enat(A3,F3)) ) ) ) )
           => aa_set3337841544853333262enat_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1051_finite__subset__induct,axiom,
    ! [F2: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(ord_less_eq_set_int(F2),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( aa_set_int_o(member_int(A3),A2)
                 => ( ~ aa_set_int_o(member_int(A3),F3)
                   => ( aa_set_int_o(P,F3)
                     => aa_set_int_o(P,insert_int(A3,F3)) ) ) ) )
           => aa_set_int_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1052_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S2: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S2)
             => ( ! [Y2: complex] :
                    ( aa_set_complex_o(member_complex(Y2),S2)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y2)),aa_complex_rat(F,X2)) )
               => ( aa_set_complex_o(P,S2)
                 => aa_set_complex_o(P,insert_complex(X2,S2)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1053_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S2: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S2)
             => ( ! [Y2: nat] :
                    ( aa_set_nat_o(member_nat(Y2),S2)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y2)),aa_nat_rat(F,X2)) )
               => ( aa_set_nat_o(P,S2)
                 => aa_set_nat_o(P,insert_nat(X2,S2)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1054_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S2: set_int] :
              ( aa_set_int_o(finite_finite_int,S2)
             => ( ! [Y2: int] :
                    ( aa_set_int_o(member_int(Y2),S2)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y2)),aa_int_rat(F,X2)) )
               => ( aa_set_int_o(P,S2)
                 => aa_set_int_o(P,insert_int(X2,S2)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1055_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,S2: set_o] :
              ( aa_set_o_o(finite_finite_o,S2)
             => ( ! [Y2: $o] :
                    ( aa_set_o_o(member_o((Y2)),S2)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Y2))),aa_o_rat(F,(X2))) )
               => ( aa_set_o_o(P,S2)
                 => aa_set_o_o(P,insert_o((X2),S2)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1056_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,S2: set_real] :
              ( aa_set_real_o(finite_finite_real,S2)
             => ( ! [Y2: real] :
                    ( aa_set_real_o(member_real(Y2),S2)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y2)),aa_real_rat(F,X2)) )
               => ( aa_set_real_o(P,S2)
                 => aa_set_real_o(P,insert_real(X2,S2)) ) ) )
         => aa_set_real_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1057_finite__ranking__induct,axiom,
    ! [S: set_Extended_enat,P: fun_se8123728892507317397enat_o,F: fun_Ex4152069839811072081at_rat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X2: extended_enat,S2: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S2)
             => ( ! [Y2: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Y2),S2)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Extended_enat_rat(F,Y2)),aa_Extended_enat_rat(F,X2)) )
               => ( aa_set3337841544853333262enat_o(P,S2)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X2,S2)) ) ) )
         => aa_set3337841544853333262enat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1058_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S2: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S2)
             => ( ! [Y2: complex] :
                    ( aa_set_complex_o(member_complex(Y2),S2)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y2)),aa_complex_num(F,X2)) )
               => ( aa_set_complex_o(P,S2)
                 => aa_set_complex_o(P,insert_complex(X2,S2)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1059_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S2: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S2)
             => ( ! [Y2: nat] :
                    ( aa_set_nat_o(member_nat(Y2),S2)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y2)),aa_nat_num(F,X2)) )
               => ( aa_set_nat_o(P,S2)
                 => aa_set_nat_o(P,insert_nat(X2,S2)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1060_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S2: set_int] :
              ( aa_set_int_o(finite_finite_int,S2)
             => ( ! [Y2: int] :
                    ( aa_set_int_o(member_int(Y2),S2)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y2)),aa_int_num(F,X2)) )
               => ( aa_set_int_o(P,S2)
                 => aa_set_int_o(P,insert_int(X2,S2)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1061_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_num] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,S2: set_o] :
              ( aa_set_o_o(finite_finite_o,S2)
             => ( ! [Y2: $o] :
                    ( aa_set_o_o(member_o((Y2)),S2)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_o_num(F,(Y2))),aa_o_num(F,(X2))) )
               => ( aa_set_o_o(P,S2)
                 => aa_set_o_o(P,insert_o((X2),S2)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1062_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_1) ) ).

% not_min_Null_member
tff(fact_1063_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_o,A2: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A5)
         => aa_set_complex_o(P,A5) )
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,F3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F3)
             => ( ~ aa_set_complex_o(member_complex(X2),F3)
               => ( aa_set_complex_o(P,F3)
                 => aa_set_complex_o(P,insert_complex(X2,F3)) ) ) )
         => aa_set_complex_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1064_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A2: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A5)
         => aa_set_nat_o(P,A5) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X2),F3)
               => ( aa_set_nat_o(P,F3)
                 => aa_set_nat_o(P,insert_nat(X2,F3)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1065_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A2: set_int] :
      ( ! [A5: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A5)
         => aa_set_int_o(P,A5) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X2),F3)
               => ( aa_set_int_o(P,F3)
                 => aa_set_int_o(P,insert_int(X2,F3)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1066_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A2: set_o] :
      ( ! [A5: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A5)
         => aa_set_o_o(P,A5) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X2)),F3)
               => ( aa_set_o_o(P,F3)
                 => aa_set_o_o(P,insert_o((X2),F3)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1067_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_o,A2: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
         => aa_set_set_nat_o(P,A5) )
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X2: set_nat,F3: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
             => ( ~ aa_set_set_nat_o(member_set_nat(X2),F3)
               => ( aa_set_set_nat_o(P,F3)
                 => aa_set_set_nat_o(P,insert_set_nat(X2,F3)) ) ) )
         => aa_set_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1068_infinite__finite__induct,axiom,
    ! [P: fun_set_real_o,A2: set_real] :
      ( ! [A5: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A5)
         => aa_set_real_o(P,A5) )
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,F3: set_real] :
              ( aa_set_real_o(finite_finite_real,F3)
             => ( ~ aa_set_real_o(member_real(X2),F3)
               => ( aa_set_real_o(P,F3)
                 => aa_set_real_o(P,insert_real(X2,F3)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1069_infinite__finite__induct,axiom,
    ! [P: fun_se8123728892507317397enat_o,A2: set_Extended_enat] :
      ( ! [A5: set_Extended_enat] :
          ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
         => aa_set3337841544853333262enat_o(P,A5) )
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X2: extended_enat,F3: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
             => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(X2),F3)
               => ( aa_set3337841544853333262enat_o(P,F3)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X2,F3)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_1070_finite__ne__induct,axiom,
    ! [F2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( ( F2 != bot_bot_set_complex )
       => ( ! [X2: complex] : aa_set_complex_o(P,insert_complex(X2,bot_bot_set_complex))
         => ( ! [X2: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( ( F3 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X2),F3)
                   => ( aa_set_complex_o(P,F3)
                     => aa_set_complex_o(P,insert_complex(X2,F3)) ) ) ) )
           => aa_set_complex_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1071_finite__ne__induct,axiom,
    ! [F2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X2: nat] : aa_set_nat_o(P,insert_nat(X2,bot_bot_set_nat))
         => ( ! [X2: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X2),F3)
                   => ( aa_set_nat_o(P,F3)
                     => aa_set_nat_o(P,insert_nat(X2,F3)) ) ) ) )
           => aa_set_nat_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1072_finite__ne__induct,axiom,
    ! [F2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X2: int] : aa_set_int_o(P,insert_int(X2,bot_bot_set_int))
         => ( ! [X2: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X2),F3)
                   => ( aa_set_int_o(P,F3)
                     => aa_set_int_o(P,insert_int(X2,F3)) ) ) ) )
           => aa_set_int_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1073_finite__ne__induct,axiom,
    ! [F2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X2: $o] : aa_set_o_o(P,insert_o((X2),bot_bot_set_o))
         => ( ! [X2: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X2)),F3)
                   => ( aa_set_o_o(P,F3)
                     => aa_set_o_o(P,insert_o((X2),F3)) ) ) ) )
           => aa_set_o_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1074_finite__ne__induct,axiom,
    ! [F2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( ( F2 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat] : aa_set_set_nat_o(P,insert_set_nat(X2,bot_bot_set_set_nat))
         => ( ! [X2: set_nat,F3: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
               => ( ( F3 != bot_bot_set_set_nat )
                 => ( ~ aa_set_set_nat_o(member_set_nat(X2),F3)
                   => ( aa_set_set_nat_o(P,F3)
                     => aa_set_set_nat_o(P,insert_set_nat(X2,F3)) ) ) ) )
           => aa_set_set_nat_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1075_finite__ne__induct,axiom,
    ! [F2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( ( F2 != bot_bot_set_real )
       => ( ! [X2: real] : aa_set_real_o(P,insert_real(X2,bot_bot_set_real))
         => ( ! [X2: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( ( F3 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real(X2),F3)
                   => ( aa_set_real_o(P,F3)
                     => aa_set_real_o(P,insert_real(X2,F3)) ) ) ) )
           => aa_set_real_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1076_finite__ne__induct,axiom,
    ! [F2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
     => ( ( F2 != bot_bo7653980558646680370d_enat )
       => ( ! [X2: extended_enat] : aa_set3337841544853333262enat_o(P,insert_Extended_enat(X2,bot_bo7653980558646680370d_enat))
         => ( ! [X2: extended_enat,F3: set_Extended_enat] :
                ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
               => ( ( F3 != bot_bo7653980558646680370d_enat )
                 => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(X2),F3)
                   => ( aa_set3337841544853333262enat_o(P,F3)
                     => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X2,F3)) ) ) ) )
           => aa_set3337841544853333262enat_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1077_finite__induct,axiom,
    ! [F2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,F3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F3)
             => ( ~ aa_set_complex_o(member_complex(X2),F3)
               => ( aa_set_complex_o(P,F3)
                 => aa_set_complex_o(P,insert_complex(X2,F3)) ) ) )
         => aa_set_complex_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1078_finite__induct,axiom,
    ! [F2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X2),F3)
               => ( aa_set_nat_o(P,F3)
                 => aa_set_nat_o(P,insert_nat(X2,F3)) ) ) )
         => aa_set_nat_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1079_finite__induct,axiom,
    ! [F2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X2),F3)
               => ( aa_set_int_o(P,F3)
                 => aa_set_int_o(P,insert_int(X2,F3)) ) ) )
         => aa_set_int_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1080_finite__induct,axiom,
    ! [F2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X2: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X2)),F3)
               => ( aa_set_o_o(P,F3)
                 => aa_set_o_o(P,insert_o((X2),F3)) ) ) )
         => aa_set_o_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1081_finite__induct,axiom,
    ! [F2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X2: set_nat,F3: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
             => ( ~ aa_set_set_nat_o(member_set_nat(X2),F3)
               => ( aa_set_set_nat_o(P,F3)
                 => aa_set_set_nat_o(P,insert_set_nat(X2,F3)) ) ) )
         => aa_set_set_nat_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1082_finite__induct,axiom,
    ! [F2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,F3: set_real] :
              ( aa_set_real_o(finite_finite_real,F3)
             => ( ~ aa_set_real_o(member_real(X2),F3)
               => ( aa_set_real_o(P,F3)
                 => aa_set_real_o(P,insert_real(X2,F3)) ) ) )
         => aa_set_real_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1083_finite__induct,axiom,
    ! [F2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X2: extended_enat,F3: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
             => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(X2),F3)
               => ( aa_set3337841544853333262enat_o(P,F3)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X2,F3)) ) ) )
         => aa_set3337841544853333262enat_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1084_zero__reorient,axiom,
    ! [Xa: literal] :
      ( ( zero_zero_literal = Xa )
    <=> ( Xa = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_1085_zero__reorient,axiom,
    ! [Xa: real] :
      ( ( zero_zero_real = Xa )
    <=> ( Xa = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1086_zero__reorient,axiom,
    ! [Xa: rat] :
      ( ( zero_zero_rat = Xa )
    <=> ( Xa = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1087_zero__reorient,axiom,
    ! [Xa: nat] :
      ( ( zero_zero_nat = Xa )
    <=> ( Xa = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1088_zero__reorient,axiom,
    ! [Xa: int] :
      ( ( zero_zero_int = Xa )
    <=> ( Xa = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1089_zero__le,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa) ).

% zero_le
tff(fact_1090_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),A)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1091_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),A)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1092_finite__has__minimal2,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((A)),A2)
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(A))
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa2)),(X2))
                 => ( (X2)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1093_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => ? [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X2),A)
            & ! [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1094_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & aa_set_int_o(ord_less_eq_set_int(X2),A)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1095_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),A)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1096_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(member_num(A),A2)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),A)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1097_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),A)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1098_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),A)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1099_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1100_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X2)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1101_finite__has__maximal2,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((A)),A2)
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(X2))
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(Xa2))
                 => ( (X2)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1102_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => ? [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),X2)
            & ! [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1103_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & aa_set_int_o(ord_less_eq_set_int(A),X2)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1104_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X2)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1105_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(member_num(A),A2)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X2)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1106_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X2)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1107_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X2)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1108_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_1109_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_1110_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_1111_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_1112_finite_OemptyI,axiom,
    aa_set_set_nat_o(finite1152437895449049373et_nat,bot_bot_set_set_nat) ).

% finite.emptyI
tff(fact_1113_finite_OemptyI,axiom,
    aa_set_real_o(finite_finite_real,bot_bot_set_real) ).

% finite.emptyI
tff(fact_1114_finite_OemptyI,axiom,
    aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,bot_bo7653980558646680370d_enat) ).

% finite.emptyI
tff(fact_1115_infinite__imp__nonempty,axiom,
    ! [S: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( S != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1116_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_1117_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_1118_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_1119_infinite__imp__nonempty,axiom,
    ! [S: set_set_nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( S != bot_bot_set_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1120_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_1121_infinite__imp__nonempty,axiom,
    ! [S: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( S != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
tff(fact_1122_rev__finite__subset,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% rev_finite_subset
tff(fact_1123_rev__finite__subset,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B)
       => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% rev_finite_subset
tff(fact_1124_rev__finite__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
       => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% rev_finite_subset
tff(fact_1125_rev__finite__subset,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B)
       => aa_set_int_o(finite_finite_int,A2) ) ) ).

% rev_finite_subset
tff(fact_1126_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_super
tff(fact_1127_infinite__super,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S),T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T2) ) ) ).

% infinite_super
tff(fact_1128_infinite__super,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S),T2)
     => ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2) ) ) ).

% infinite_super
tff(fact_1129_infinite__super,axiom,
    ! [S: set_int,T2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S),T2)
     => ( ~ aa_set_int_o(finite_finite_int,S)
       => ~ aa_set_int_o(finite_finite_int,T2) ) ) ).

% infinite_super
tff(fact_1130_finite__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_subset
tff(fact_1131_finite__subset,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B)
       => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% finite_subset
tff(fact_1132_finite__subset,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
       => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% finite_subset
tff(fact_1133_finite__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(finite_finite_int,B)
       => aa_set_int_o(finite_finite_int,A2) ) ) ).

% finite_subset
tff(fact_1134_finite_OinsertI,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => aa_set_o_o(finite_finite_o,insert_o((A),A2)) ) ).

% finite.insertI
tff(fact_1135_finite_OinsertI,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,insert_set_nat(A,A2)) ) ).

% finite.insertI
tff(fact_1136_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => aa_set_real_o(finite_finite_real,insert_real(A,A2)) ) ).

% finite.insertI
tff(fact_1137_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_nat_o(finite_finite_nat,insert_nat(A,A2)) ) ).

% finite.insertI
tff(fact_1138_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => aa_set_int_o(finite_finite_int,insert_int(A,A2)) ) ).

% finite.insertI
tff(fact_1139_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A2)) ) ).

% finite.insertI
tff(fact_1140_finite_OinsertI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,insert_Extended_enat(A,A2)) ) ).

% finite.insertI
tff(fact_1141_finite__UnI,axiom,
    ! [F2: set_complex,G: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,G)
       => aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(F2,G)) ) ) ).

% finite_UnI
tff(fact_1142_finite__UnI,axiom,
    ! [F2: set_Extended_enat,G: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,G)
       => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(F2,G)) ) ) ).

% finite_UnI
tff(fact_1143_finite__UnI,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(finite_finite_nat,G)
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F2),G)) ) ) ).

% finite_UnI
tff(fact_1144_finite__UnI,axiom,
    ! [F2: set_o,G: set_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(finite_finite_o,G)
       => aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,F2),G)) ) ) ).

% finite_UnI
tff(fact_1145_finite__UnI,axiom,
    ! [F2: set_int,G: set_int] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(finite_finite_int,G)
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F2),G)) ) ) ).

% finite_UnI
tff(fact_1146_Un__infinite,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(S,T2)) ) ).

% Un_infinite
tff(fact_1147_Un__infinite,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(S,T2)) ) ).

% Un_infinite
tff(fact_1148_Un__infinite,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)) ) ).

% Un_infinite
tff(fact_1149_Un__infinite,axiom,
    ! [S: set_o,T2: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ~ aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),T2)) ) ).

% Un_infinite
tff(fact_1150_Un__infinite,axiom,
    ! [S: set_int,T2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2)) ) ).

% Un_infinite
tff(fact_1151_infinite__Un,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(S,T2))
    <=> ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
        | ~ aa_set_complex_o(finite3207457112153483333omplex,T2) ) ) ).

% infinite_Un
tff(fact_1152_infinite__Un,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(S,T2))
    <=> ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
        | ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2) ) ) ).

% infinite_Un
tff(fact_1153_infinite__Un,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2))
    <=> ( ~ aa_set_nat_o(finite_finite_nat,S)
        | ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_Un
tff(fact_1154_infinite__Un,axiom,
    ! [S: set_o,T2: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),T2))
    <=> ( ~ aa_set_o_o(finite_finite_o,S)
        | ~ aa_set_o_o(finite_finite_o,T2) ) ) ).

% infinite_Un
tff(fact_1155_infinite__Un,axiom,
    ! [S: set_int,T2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2))
    <=> ( ~ aa_set_int_o(finite_finite_int,S)
        | ~ aa_set_int_o(finite_finite_int,T2) ) ) ).

% infinite_Un
tff(fact_1156_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(Xa2))
                 => ( (X2)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1157_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1158_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1159_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
            & ! [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1160_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1161_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1162_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1163_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1164_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1165_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa2)),(X2))
                 => ( (X2)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1166_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1167_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1168_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
            & ! [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1169_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1170_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A2)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1171_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A2)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1172_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1173_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X2)
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1174_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A5: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A5)
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A5) ) ) ) ).

% finite.cases
tff(fact_1175_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A5: set_nat] :
              ( ? [A3: nat] : A = insert_nat(A3,A5)
             => ~ aa_set_nat_o(finite_finite_nat,A5) ) ) ) ).

% finite.cases
tff(fact_1176_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A5: set_int] :
              ( ? [A3: int] : A = insert_int(A3,A5)
             => ~ aa_set_int_o(finite_finite_int,A5) ) ) ) ).

% finite.cases
tff(fact_1177_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A5: set_o] :
              ( ? [A3: $o] : A = insert_o((A3),A5)
             => ~ aa_set_o_o(finite_finite_o,A5) ) ) ) ).

% finite.cases
tff(fact_1178_finite_Ocases,axiom,
    ! [A: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A)
     => ( ( A != bot_bot_set_set_nat )
       => ~ ! [A5: set_set_nat] :
              ( ? [A3: set_nat] : A = insert_set_nat(A3,A5)
             => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A5) ) ) ) ).

% finite.cases
tff(fact_1179_finite_Ocases,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ~ ! [A5: set_real] :
              ( ? [A3: real] : A = insert_real(A3,A5)
             => ~ aa_set_real_o(finite_finite_real,A5) ) ) ) ).

% finite.cases
tff(fact_1180_finite_Ocases,axiom,
    ! [A: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A)
     => ( ( A != bot_bo7653980558646680370d_enat )
       => ~ ! [A5: set_Extended_enat] :
              ( ? [A3: extended_enat] : A = insert_Extended_enat(A3,A5)
             => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5) ) ) ) ).

% finite.cases
tff(fact_1181_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
    <=> ( ( A = bot_bot_set_complex )
        | ? [A6: set_complex,A7: complex] :
            ( ( A = insert_complex(A7,A6) )
            & aa_set_complex_o(finite3207457112153483333omplex,A6) ) ) ) ).

% finite.simps
tff(fact_1182_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A6: set_nat,A7: nat] :
            ( ( A = insert_nat(A7,A6) )
            & aa_set_nat_o(finite_finite_nat,A6) ) ) ) ).

% finite.simps
tff(fact_1183_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A6: set_int,A7: int] :
            ( ( A = insert_int(A7,A6) )
            & aa_set_int_o(finite_finite_int,A6) ) ) ) ).

% finite.simps
tff(fact_1184_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A6: set_o,A7: $o] :
            ( ( A = insert_o((A7),A6) )
            & aa_set_o_o(finite_finite_o,A6) ) ) ) ).

% finite.simps
tff(fact_1185_finite_Osimps,axiom,
    ! [A: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A)
    <=> ( ( A = bot_bot_set_set_nat )
        | ? [A6: set_set_nat,A7: set_nat] :
            ( ( A = insert_set_nat(A7,A6) )
            & aa_set_set_nat_o(finite1152437895449049373et_nat,A6) ) ) ) ).

% finite.simps
tff(fact_1186_finite_Osimps,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
    <=> ( ( A = bot_bot_set_real )
        | ? [A6: set_real,A7: real] :
            ( ( A = insert_real(A7,A6) )
            & aa_set_real_o(finite_finite_real,A6) ) ) ) ).

% finite.simps
tff(fact_1187_finite_Osimps,axiom,
    ! [A: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A)
    <=> ( ( A = bot_bo7653980558646680370d_enat )
        | ? [A6: set_Extended_enat,A7: extended_enat] :
            ( ( A = insert_Extended_enat(A7,A6) )
            & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6) ) ) ) ).

% finite.simps
tff(fact_1188_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Na)),Na) ) ).

% buildup_gives_valid
tff(fact_1189_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_1190_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_1191_arg__min__least,axiom,
    ! [S: set_complex,Ya: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ( S != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Ya),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S))),aa_complex_rat(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1192_arg__min__least,axiom,
    ! [S: set_nat,Ya: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Ya),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,lattic6811802900495863747at_rat(F,S))),aa_nat_rat(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1193_arg__min__least,axiom,
    ! [S: set_int,Ya: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Ya),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S))),aa_int_rat(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1194_arg__min__least,axiom,
    ! [S: set_o,Ya: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Ya)),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,aa_set_o_o(lattic2140725968369957399_o_rat(F),S))),aa_o_rat(F,(Ya))) ) ) ) ).

% arg_min_least
tff(fact_1195_arg__min__least,axiom,
    ! [S: set_real,Ya: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ( S != bot_bot_set_real )
       => ( aa_set_real_o(member_real(Ya),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S))),aa_real_rat(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1196_arg__min__least,axiom,
    ! [S: set_Extended_enat,Ya: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ya),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Extended_enat_rat(F,lattic3210252021154270693at_rat(F,S))),aa_Extended_enat_rat(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1197_arg__min__least,axiom,
    ! [S: set_complex,Ya: complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ( S != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Ya),S)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S))),aa_complex_num(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1198_arg__min__least,axiom,
    ! [S: set_nat,Ya: nat,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Ya),S)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,lattic4004264746738138117at_num(F,S))),aa_nat_num(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1199_arg__min__least,axiom,
    ! [S: set_int,Ya: int,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Ya),S)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S))),aa_int_num(F,Ya)) ) ) ) ).

% arg_min_least
tff(fact_1200_arg__min__least,axiom,
    ! [S: set_o,Ya: $o,F: fun_o_num] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Ya)),S)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_o_num(F,aa_set_o_o(lattic8556559851467007577_o_num(F),S))),aa_o_num(F,(Ya))) ) ) ) ).

% arg_min_least
tff(fact_1201_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X2: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R,X2),X2)
       => ( ! [X2: complex,Y: complex,Z2: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R,X2),Y)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R,Y),Z2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,X2),Z2) ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),A2)
               => ? [Y2: complex] :
                    ( aa_set_complex_o(member_complex(Y2),A2)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R,X2),Y2) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1202_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z2) ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A2)
               => ? [Y2: nat] :
                    ( aa_set_nat_o(member_nat(Y2),A2)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1203_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X2: int] : ~ aa_int_o(aa_int_fun_int_o(R,X2),X2)
       => ( ! [X2: int,Y: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X2),Y)
             => ( aa_int_o(aa_int_fun_int_o(R,Y),Z2)
               => aa_int_o(aa_int_fun_int_o(R,X2),Z2) ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),A2)
               => ? [Y2: int] :
                    ( aa_set_int_o(member_int(Y2),A2)
                    & aa_int_o(aa_int_fun_int_o(R,X2),Y2) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1204_finite__transitivity__chain,axiom,
    ! [A2: set_o,R: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [X2: $o] : ~ aa_o_o(aa_o_fun_o_o(R,(X2)),(X2))
       => ( ! [X2: $o,Y: $o,Z2: $o] :
              ( aa_o_o(aa_o_fun_o_o(R,(X2)),(Y))
             => ( aa_o_o(aa_o_fun_o_o(R,(Y)),(Z2))
               => aa_o_o(aa_o_fun_o_o(R,(X2)),(Z2)) ) )
         => ( ! [X2: $o] :
                ( aa_set_o_o(member_o((X2)),A2)
               => ? [Y2: $o] :
                    ( aa_set_o_o(member_o((Y2)),A2)
                    & aa_o_o(aa_o_fun_o_o(R,(X2)),(Y2)) ) )
           => ( A2 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1205_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se2170385732985038782_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ! [X2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X2),X2)
       => ( ! [X2: set_nat,Y: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X2),Y)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,Y),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X2),Z2) ) )
         => ( ! [X2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(X2),A2)
               => ? [Y2: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat(Y2),A2)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X2),Y2) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1206_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X2: real] : ~ aa_real_o(aa_real_fun_real_o(R,X2),X2)
       => ( ! [X2: real,Y: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(R,X2),Y)
             => ( aa_real_o(aa_real_fun_real_o(R,Y),Z2)
               => aa_real_o(aa_real_fun_real_o(R,X2),Z2) ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),A2)
               => ? [Y2: real] :
                    ( aa_set_real_o(member_real(Y2),A2)
                    & aa_real_o(aa_real_fun_real_o(R,X2),Y2) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1207_finite__transitivity__chain,axiom,
    ! [A2: set_Extended_enat,R: fun_Ex239764483819389678enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [X2: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X2),X2)
       => ( ! [X2: extended_enat,Y: extended_enat,Z2: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X2),Y)
             => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,Y),Z2)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X2),Z2) ) )
         => ( ! [X2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
               => ? [Y2: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Y2),A2)
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X2),Y2) ) )
           => ( A2 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1208_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% deg_not_0
tff(fact_1209_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_1210_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_1211_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_1212_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_1213_Sup__fin_Oinsert,axiom,
    ! [A2: set_set_o,Xa: set_o] :
      ( finite_finite_set_o(A2)
     => ( ( A2 != bot_bot_set_set_o )
       => ( lattic3158155371183623599_set_o(insert_set_o(Xa,A2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),lattic3158155371183623599_set_o(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1214_Sup__fin_Oinsert,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( lattic8880645941091133547et_int(insert_set_int(Xa,A2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),lattic8880645941091133547et_int(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1215_Sup__fin_Oinsert,axiom,
    ! [A2: set_filter_nat,Xa: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ( lattic5930898082463196905er_nat(insert_filter_nat(Xa,A2)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),lattic5930898082463196905er_nat(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1216_Sup__fin_Oinsert,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( lattic1093996805478795353in_nat(insert_nat(Xa,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),lattic1093996805478795353in_nat(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1217_Sup__fin_Oinsert,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( lattic1091506334969745077in_int(insert_int(Xa,A2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),lattic1091506334969745077in_int(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1218_Sup__fin_Oinsert,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((Xa),A2))
        <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(Xa)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1219_Sup__fin_Oinsert,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( lattic3835124923745554447et_nat(insert_set_nat(Xa,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),lattic3835124923745554447et_nat(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1220_Sup__fin_Oinsert,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( lattic8928443293348198069n_real(insert_real(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,Xa),lattic8928443293348198069n_real(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1221_Sup__fin_Oinsert,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( lattic5005175426920976669d_enat(insert_Extended_enat(Xa,A2)) = sup_su3973961784419623482d_enat(Xa,lattic5005175426920976669d_enat(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1222_Leaf__0__not,axiom,
    ! [A: $o,B2: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B2))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_1223_Sup__fin_Ounion,axiom,
    ! [A2: set_set_o,B: set_set_o] :
      ( finite_finite_set_o(A2)
     => ( ( A2 != bot_bot_set_set_o )
       => ( finite_finite_set_o(B)
         => ( ( B != bot_bot_set_set_o )
           => ( lattic3158155371183623599_set_o(sup_sup_set_set_o(A2,B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,lattic3158155371183623599_set_o(A2)),lattic3158155371183623599_set_o(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1224_Sup__fin_Ounion,axiom,
    ! [A2: set_set_int,B: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( finite6197958912794628473et_int(B)
         => ( ( B != bot_bot_set_set_int )
           => ( lattic8880645941091133547et_int(sup_sup_set_set_int(A2,B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,lattic8880645941091133547et_int(A2)),lattic8880645941091133547et_int(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1225_Sup__fin_Ounion,axiom,
    ! [A2: set_filter_nat,B: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ( finite2119507909894593271er_nat(B)
         => ( ( B != bot_bo498966703094740906er_nat )
           => ( lattic5930898082463196905er_nat(sup_su3181624671392095810er_nat(A2,B)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,lattic5930898082463196905er_nat(A2)),lattic5930898082463196905er_nat(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1226_Sup__fin_Ounion,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B)
         => ( ( B != bot_bot_set_nat )
           => ( lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic1093996805478795353in_nat(A2)),lattic1093996805478795353in_nat(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1227_Sup__fin_Ounion,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B)
         => ( ( B != bot_bot_set_int )
           => ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic1091506334969745077in_int(A2)),lattic1091506334969745077in_int(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1228_Sup__fin_Ounion,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B)
         => ( ( B != bot_bot_set_o )
           => ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
            <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),aa_set_o_o(lattic1508158080041050831_fin_o,B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1229_Sup__fin_Ounion,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
         => ( ( B != bot_bot_set_set_nat )
           => ( lattic3835124923745554447et_nat(sup_sup_set_set_nat(A2,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(A2)),lattic3835124923745554447et_nat(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1230_Sup__fin_Ounion,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_real_o(finite_finite_real,B)
         => ( ( B != bot_bot_set_real )
           => ( lattic8928443293348198069n_real(sup_sup_set_real(A2,B)) = aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,lattic8928443293348198069n_real(A2)),lattic8928443293348198069n_real(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1231_Sup__fin_Ounion,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
         => ( ( B != bot_bo7653980558646680370d_enat )
           => ( lattic5005175426920976669d_enat(sup_su4489774667511045786d_enat(A2,B)) = sup_su3973961784419623482d_enat(lattic5005175426920976669d_enat(A2),lattic5005175426920976669d_enat(B)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1232_not__gr__zero,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_1233_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1234_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_1235_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_1236_Sup__fin_Osingleton,axiom,
    ! [Xa: nat] : lattic1093996805478795353in_nat(insert_nat(Xa,bot_bot_set_nat)) = Xa ).

% Sup_fin.singleton
tff(fact_1237_Sup__fin_Osingleton,axiom,
    ! [Xa: int] : lattic1091506334969745077in_int(insert_int(Xa,bot_bot_set_int)) = Xa ).

% Sup_fin.singleton
tff(fact_1238_Sup__fin_Osingleton,axiom,
    ! [Xa: $o] :
      ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((Xa),bot_bot_set_o))
    <=> (Xa) ) ).

% Sup_fin.singleton
tff(fact_1239_Sup__fin_Osingleton,axiom,
    ! [Xa: set_nat] : lattic3835124923745554447et_nat(insert_set_nat(Xa,bot_bot_set_set_nat)) = Xa ).

% Sup_fin.singleton
tff(fact_1240_Sup__fin_Osingleton,axiom,
    ! [Xa: real] : lattic8928443293348198069n_real(insert_real(Xa,bot_bot_set_real)) = Xa ).

% Sup_fin.singleton
tff(fact_1241_Sup__fin_Osingleton,axiom,
    ! [Xa: extended_enat] : lattic5005175426920976669d_enat(insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = Xa ).

% Sup_fin.singleton
tff(fact_1242_order__less__imp__not__less,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ).

% order_less_imp_not_less
tff(fact_1243_order__less__imp__not__less,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ).

% order_less_imp_not_less
tff(fact_1244_order__less__imp__not__less,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ).

% order_less_imp_not_less
tff(fact_1245_order__less__imp__not__less,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ).

% order_less_imp_not_less
tff(fact_1246_order__less__imp__not__less,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ).

% order_less_imp_not_less
tff(fact_1247_order__less__imp__not__eq2,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( Ya != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1248_order__less__imp__not__eq2,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( Ya != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1249_order__less__imp__not__eq2,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ( Ya != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1250_order__less__imp__not__eq2,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ( Ya != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1251_order__less__imp__not__eq2,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ( Ya != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_1252_order__less__imp__not__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( Xa != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_1253_order__less__imp__not__eq,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( Xa != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_1254_order__less__imp__not__eq,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ( Xa != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_1255_order__less__imp__not__eq,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ( Xa != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_1256_order__less__imp__not__eq,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ( Xa != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_1257_linorder__less__linear,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
      | ( Xa = Ya )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ).

% linorder_less_linear
tff(fact_1258_linorder__less__linear,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
      | ( Xa = Ya )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ).

% linorder_less_linear
tff(fact_1259_linorder__less__linear,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
      | ( Xa = Ya )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ).

% linorder_less_linear
tff(fact_1260_linorder__less__linear,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
      | ( Xa = Ya )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ).

% linorder_less_linear
tff(fact_1261_linorder__less__linear,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
      | ( Xa = Ya )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ).

% linorder_less_linear
tff(fact_1262_order__less__imp__triv,axiom,
    ! [Xa: real,Ya: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1263_order__less__imp__triv,axiom,
    ! [Xa: rat,Ya: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1264_order__less__imp__triv,axiom,
    ! [Xa: num,Ya: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1265_order__less__imp__triv,axiom,
    ! [Xa: nat,Ya: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1266_order__less__imp__triv,axiom,
    ! [Xa: int,Ya: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1267_order__less__not__sym,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ).

% order_less_not_sym
tff(fact_1268_order__less__not__sym,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ).

% order_less_not_sym
tff(fact_1269_order__less__not__sym,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ).

% order_less_not_sym
tff(fact_1270_order__less__not__sym,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ).

% order_less_not_sym
tff(fact_1271_order__less__not__sym,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ).

% order_less_not_sym
tff(fact_1272_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1273_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1274_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1275_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1276_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1277_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1278_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1279_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1280_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1281_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_1282_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1283_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1284_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B2: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1285_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1286_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1287_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B2: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1288_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1289_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1290_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1291_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B2: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_1292_order__less__irrefl,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Xa) ).

% order_less_irrefl
tff(fact_1293_order__less__irrefl,axiom,
    ! [Xa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Xa) ).

% order_less_irrefl
tff(fact_1294_order__less__irrefl,axiom,
    ! [Xa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Xa) ).

% order_less_irrefl
tff(fact_1295_order__less__irrefl,axiom,
    ! [Xa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Xa) ).

% order_less_irrefl
tff(fact_1296_order__less__irrefl,axiom,
    ! [Xa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xa) ).

% order_less_irrefl
tff(fact_1297_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_real(F,B2) = C2 )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1298_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_rat(F,B2) = C2 )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1299_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_num(F,B2) = C2 )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1300_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_nat(F,B2) = C2 )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1301_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_int(F,B2) = C2 )
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1302_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_real(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1303_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_rat(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1304_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_num(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1305_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_nat(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1306_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_int(F,B2) = C2 )
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_1307_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B2: real,C2: real] :
      ( ( A = aa_real_real(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1308_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B2: real,C2: real] :
      ( ( A = aa_real_rat(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1309_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B2: real,C2: real] :
      ( ( A = aa_real_num(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1310_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C2: real] :
      ( ( A = aa_real_nat(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1311_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B2: real,C2: real] :
      ( ( A = aa_real_int(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1312_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( ( A = aa_rat_real(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1313_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1314_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1315_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1316_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B2: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1317_order__less__trans,axiom,
    ! [Xa: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1318_order__less__trans,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),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,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1319_order__less__trans,axiom,
    ! [Xa: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),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,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1320_order__less__trans,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),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,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1321_order__less__trans,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),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,Xa),Z) ) ) ).

% order_less_trans
tff(fact_1322_order__less__asym_H,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ).

% order_less_asym'
tff(fact_1323_order__less__asym_H,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ).

% order_less_asym'
tff(fact_1324_order__less__asym_H,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A) ) ).

% order_less_asym'
tff(fact_1325_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A) ) ).

% order_less_asym'
tff(fact_1326_order__less__asym_H,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ).

% order_less_asym'
tff(fact_1327_linorder__neq__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( ( Xa != Ya )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1328_linorder__neq__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( Xa != Ya )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1329_linorder__neq__iff,axiom,
    ! [Xa: num,Ya: num] :
      ( ( Xa != Ya )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1330_linorder__neq__iff,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( Xa != Ya )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1331_linorder__neq__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( ( Xa != Ya )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ) ).

% linorder_neq_iff
tff(fact_1332_order__less__asym,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ).

% order_less_asym
tff(fact_1333_order__less__asym,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ).

% order_less_asym
tff(fact_1334_order__less__asym,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ).

% order_less_asym
tff(fact_1335_order__less__asym,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ).

% order_less_asym
tff(fact_1336_order__less__asym,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ).

% order_less_asym
tff(fact_1337_linorder__neqE,axiom,
    ! [Xa: real,Ya: real] :
      ( ( Xa != Ya )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ).

% linorder_neqE
tff(fact_1338_linorder__neqE,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( Xa != Ya )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ) ).

% linorder_neqE
tff(fact_1339_linorder__neqE,axiom,
    ! [Xa: num,Ya: num] :
      ( ( Xa != Ya )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ) ).

% linorder_neqE
tff(fact_1340_linorder__neqE,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( Xa != Ya )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ) ).

% linorder_neqE
tff(fact_1341_linorder__neqE,axiom,
    ! [Xa: int,Ya: int] :
      ( ( Xa != Ya )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ) ).

% linorder_neqE
tff(fact_1342_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1343_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1344_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1345_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1346_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1347_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_1348_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_1349_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_1350_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_1351_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_1352_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans
tff(fact_1353_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans
tff(fact_1354_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans
tff(fact_1355_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans
tff(fact_1356_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans
tff(fact_1357_not__less__iff__gr__or__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa)
        | ( Xa = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1358_not__less__iff__gr__or__eq,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa)
        | ( Xa = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1359_not__less__iff__gr__or__eq,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa)
        | ( Xa = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1360_not__less__iff__gr__or__eq,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa)
        | ( Xa = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1361_not__less__iff__gr__or__eq,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa)
        | ( Xa = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1362_order_Ostrict__trans,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),C2) ) ) ).

% order.strict_trans
tff(fact_1363_order_Ostrict__trans,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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,A),C2) ) ) ).

% order.strict_trans
tff(fact_1364_order_Ostrict__trans,axiom,
    ! [A: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),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,A),C2) ) ) ).

% order.strict_trans
tff(fact_1365_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans
tff(fact_1366_order_Ostrict__trans,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans
tff(fact_1367_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B2: real] :
      ( ! [A3: real,B4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B4)
         => aa_real_o(aa_real_fun_real_o(P,A3),B4) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P,A3),A3)
       => ( ! [A3: real,B4: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B4),A3)
             => aa_real_o(aa_real_fun_real_o(P,A3),B4) )
         => aa_real_o(aa_real_fun_real_o(P,A),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_1368_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B2: rat] :
      ( ! [A3: rat,B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B4)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B4) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A3),A3)
       => ( ! [A3: rat,B4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B4),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P,A3),B4) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_1369_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B2: num] :
      ( ! [A3: num,B4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B4)
         => aa_num_o(aa_num_fun_num_o(P,A3),B4) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P,A3),A3)
       => ( ! [A3: num,B4: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B4),A3)
             => aa_num_o(aa_num_fun_num_o(P,A3),B4) )
         => aa_num_o(aa_num_fun_num_o(P,A),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_1370_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B2: nat] :
      ( ! [A3: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B4)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B4) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
       => ( ! [A3: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B4),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),B4) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_1371_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B2: int] :
      ( ! [A3: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B4)
         => aa_int_o(aa_int_fun_int_o(P,A3),B4) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
       => ( ! [A3: int,B4: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B4),A3)
             => aa_int_o(aa_int_fun_int_o(P,A3),B4) )
         => aa_int_o(aa_int_fun_int_o(P,A),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_1372_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N: nat] :
          ( aa_nat_o(P,N)
          & ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
             => ~ aa_nat_o(P,M) ) ) ) ).

% exists_least_iff
tff(fact_1373_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_1374_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_1375_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_1376_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_1377_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_1378_dual__order_Oasym,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% dual_order.asym
tff(fact_1379_dual__order_Oasym,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% dual_order.asym
tff(fact_1380_dual__order_Oasym,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2) ) ).

% dual_order.asym
tff(fact_1381_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% dual_order.asym
tff(fact_1382_dual__order_Oasym,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% dual_order.asym
tff(fact_1383_linorder__cases,axiom,
    ! [Xa: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( ( Xa != Ya )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ).

% linorder_cases
tff(fact_1384_linorder__cases,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( ( Xa != Ya )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ) ).

% linorder_cases
tff(fact_1385_linorder__cases,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ( ( Xa != Ya )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ) ).

% linorder_cases
tff(fact_1386_linorder__cases,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ( ( Xa != Ya )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ) ).

% linorder_cases
tff(fact_1387_linorder__cases,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ( ( Xa != Ya )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ) ).

% linorder_cases
tff(fact_1388_antisym__conv3,axiom,
    ! [Ya: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv3
tff(fact_1389_antisym__conv3,axiom,
    ! [Ya: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv3
tff(fact_1390_antisym__conv3,axiom,
    ! [Ya: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv3
tff(fact_1391_antisym__conv3,axiom,
    ! [Ya: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv3
tff(fact_1392_antisym__conv3,axiom,
    ! [Ya: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv3
tff(fact_1393_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X2)
             => aa_nat_o(P,Y2) )
         => aa_nat_o(P,X2) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_1394_ord__less__eq__trans,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( B2 = C2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_1395_ord__less__eq__trans,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( B2 = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_1396_ord__less__eq__trans,axiom,
    ! [A: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( ( B2 = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_1397_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( ( B2 = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_1398_ord__less__eq__trans,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( ( B2 = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_1399_ord__eq__less__trans,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_1400_ord__eq__less__trans,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_1401_ord__eq__less__trans,axiom,
    ! [A: num,B2: num,C2: num] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_1402_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_1403_ord__eq__less__trans,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( A = 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,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_1404_order_Oasym,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ).

% order.asym
tff(fact_1405_order_Oasym,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ).

% order.asym
tff(fact_1406_order_Oasym,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A) ) ).

% order.asym
tff(fact_1407_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A) ) ).

% order.asym
tff(fact_1408_order_Oasym,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ).

% order.asym
tff(fact_1409_less__imp__neq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( Xa != Ya ) ) ).

% less_imp_neq
tff(fact_1410_less__imp__neq,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( Xa != Ya ) ) ).

% less_imp_neq
tff(fact_1411_less__imp__neq,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ( Xa != Ya ) ) ).

% less_imp_neq
tff(fact_1412_less__imp__neq,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ( Xa != Ya ) ) ).

% less_imp_neq
tff(fact_1413_less__imp__neq,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ( Xa != Ya ) ) ).

% less_imp_neq
tff(fact_1414_dense,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Ya) ) ) ).

% dense
tff(fact_1415_dense,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Ya) ) ) ).

% dense
tff(fact_1416_gt__ex,axiom,
    ! [Xa: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X_1) ).

% gt_ex
tff(fact_1417_gt__ex,axiom,
    ! [Xa: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X_1) ).

% gt_ex
tff(fact_1418_gt__ex,axiom,
    ! [Xa: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X_1) ).

% gt_ex
tff(fact_1419_gt__ex,axiom,
    ! [Xa: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X_1) ).

% gt_ex
tff(fact_1420_lt__ex,axiom,
    ! [Xa: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ).

% lt_ex
tff(fact_1421_lt__ex,axiom,
    ! [Xa: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ).

% lt_ex
tff(fact_1422_lt__ex,axiom,
    ! [Xa: int] :
    ? [Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ).

% lt_ex
tff(fact_1423_nat__neq__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( M2 != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% nat_neq_iff
tff(fact_1424_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_1425_less__not__refl2,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
     => ( M2 != Na ) ) ).

% less_not_refl2
tff(fact_1426_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_1427_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_1428_nat__less__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
             => aa_nat_o(P,M3) )
         => aa_nat_o(P,N2) )
     => aa_nat_o(P,Na) ) ).

% nat_less_induct
tff(fact_1429_infinite__descent,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(P,N2)
         => ? [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
              & ~ aa_nat_o(P,M3) ) )
     => aa_nat_o(P,Na) ) ).

% infinite_descent
tff(fact_1430_linorder__neqE__nat,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( Xa != Ya )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_1431_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_1432_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_1433_eq__imp__le,axiom,
    ! [M2: nat,Na: nat] :
      ( ( M2 = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% eq_imp_le
tff(fact_1434_le__antisym,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( M2 = Na ) ) ) ).

% le_antisym
tff(fact_1435_nat__less__le,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
        & ( M2 != Na ) ) ) ).

% nat_less_le
tff(fact_1436_nat__le__linear,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ) ).

% nat_le_linear
tff(fact_1437_less__imp__le__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% less_imp_le_nat
tff(fact_1438_le__eq__less__or__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | ( M2 = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_1439_less__or__eq__imp__le,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | ( M2 = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% less_or_eq_imp_le
tff(fact_1440_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) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y2: nat] :
                ( aa_nat_o(P,Y2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_1441_le__neq__implies__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( ( M2 != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% le_neq_implies_less
tff(fact_1442_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_1443_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_1444_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_1445_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_1446_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_1447_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_1448_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_1449_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_1450_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1451_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_1452_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_1453_gr__implies__not0,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1454_infinite__descent0,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( ~ aa_nat_o(P,N2)
             => ? [M3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
                  & ~ aa_nat_o(P,M3) ) ) )
       => aa_nat_o(P,Na) ) ) ).

% infinite_descent0
tff(fact_1455_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M4)
         => ? [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N3)
              & aa_set_nat_o(member_nat(N3),S) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S) ) ).

% unbounded_k_infinite
tff(fact_1456_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
          & aa_set_nat_o(member_nat(N),S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1457_leD,axiom,
    ! [Ya: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% leD
tff(fact_1458_leD,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
     => ~ aa_set_int_o(ord_less_set_int(Xa),Ya) ) ).

% leD
tff(fact_1459_leD,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya) ) ).

% leD
tff(fact_1460_leD,axiom,
    ! [Ya: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya) ) ).

% leD
tff(fact_1461_leD,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ).

% leD
tff(fact_1462_leD,axiom,
    ! [Ya: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya) ) ).

% leD
tff(fact_1463_leI,axiom,
    ! [Xa: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa) ) ).

% leI
tff(fact_1464_leI,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa) ) ).

% leI
tff(fact_1465_leI,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa) ) ).

% leI
tff(fact_1466_leI,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa) ) ).

% leI
tff(fact_1467_leI,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa) ) ).

% leI
tff(fact_1468_nless__le,axiom,
    ! [A: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_1469_nless__le,axiom,
    ! [A: set_int,B2: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(A),B2)
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_1470_nless__le,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_1471_nless__le,axiom,
    ! [A: num,B2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_1472_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_1473_nless__le,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_1474_antisym__conv1,axiom,
    ! [Xa: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv1
tff(fact_1475_antisym__conv1,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv1
tff(fact_1476_antisym__conv1,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv1
tff(fact_1477_antisym__conv1,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv1
tff(fact_1478_antisym__conv1,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv1
tff(fact_1479_antisym__conv1,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv1
tff(fact_1480_antisym__conv2,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv2
tff(fact_1481_antisym__conv2,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( ~ aa_set_int_o(ord_less_set_int(Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv2
tff(fact_1482_antisym__conv2,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv2
tff(fact_1483_antisym__conv2,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv2
tff(fact_1484_antisym__conv2,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv2
tff(fact_1485_antisym__conv2,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
      <=> ( Xa = Ya ) ) ) ).

% antisym_conv2
tff(fact_1486_dense__ge,axiom,
    ! [Z: real,Ya: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),X2) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ).

% dense_ge
tff(fact_1487_dense__ge,axiom,
    ! [Z: rat,Ya: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),X2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ).

% dense_ge
tff(fact_1488_dense__le,axiom,
    ! [Ya: real,Z: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ).

% dense_le
tff(fact_1489_dense__le,axiom,
    ! [Ya: rat,Z: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Ya)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ).

% dense_le
tff(fact_1490_less__le__not__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa) ) ) ).

% less_le_not_le
tff(fact_1491_less__le__not__le,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Ya)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
        & ~ aa_set_int_o(ord_less_eq_set_int(Ya),Xa) ) ) ).

% less_le_not_le
tff(fact_1492_less__le__not__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa) ) ) ).

% less_le_not_le
tff(fact_1493_less__le__not__le,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa) ) ) ).

% less_le_not_le
tff(fact_1494_less__le__not__le,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa) ) ) ).

% less_le_not_le
tff(fact_1495_less__le__not__le,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa) ) ) ).

% less_le_not_le
tff(fact_1496_not__le__imp__less,axiom,
    ! [Ya: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% not_le_imp_less
tff(fact_1497_not__le__imp__less,axiom,
    ! [Ya: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya) ) ).

% not_le_imp_less
tff(fact_1498_not__le__imp__less,axiom,
    ! [Ya: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya) ) ).

% not_le_imp_less
tff(fact_1499_not__le__imp__less,axiom,
    ! [Ya: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ).

% not_le_imp_less
tff(fact_1500_not__le__imp__less,axiom,
    ! [Ya: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya) ) ).

% not_le_imp_less
tff(fact_1501_order_Oorder__iff__strict,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_1502_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ( aa_set_int_o(ord_less_set_int(A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_1503_order_Oorder__iff__strict,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_1504_order_Oorder__iff__strict,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_1505_order_Oorder__iff__strict,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_1506_order_Oorder__iff__strict,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_1507_order_Ostrict__iff__order,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_1508_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_1509_order_Ostrict__iff__order,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_1510_order_Ostrict__iff__order,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_1511_order_Ostrict__iff__order,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_1512_order_Ostrict__iff__order,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_1513_order_Ostrict__trans1,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1514_order_Ostrict__trans1,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_set_int(B2),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% order.strict_trans1
tff(fact_1515_order_Ostrict__trans1,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1516_order_Ostrict__trans1,axiom,
    ! [A: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),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,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1517_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1518_order_Ostrict__trans1,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans1
tff(fact_1519_order_Ostrict__trans2,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1520_order_Ostrict__trans2,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% order.strict_trans2
tff(fact_1521_order_Ostrict__trans2,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1522_order_Ostrict__trans2,axiom,
    ! [A: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),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,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1523_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),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,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1524_order_Ostrict__trans2,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),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,A),C2) ) ) ).

% order.strict_trans2
tff(fact_1525_order_Ostrict__iff__not,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_1526_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & ~ aa_set_int_o(ord_less_eq_set_int(B2),A) ) ) ).

% order.strict_iff_not
tff(fact_1527_order_Ostrict__iff__not,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_1528_order_Ostrict__iff__not,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_1529_order_Ostrict__iff__not,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_1530_order_Ostrict__iff__not,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_1531_dense__ge__bounded,axiom,
    ! [Z: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Xa)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),W) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ) ).

% dense_ge_bounded
tff(fact_1532_dense__ge__bounded,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Xa)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ) ).

% dense_ge_bounded
tff(fact_1533_dense__le__bounded,axiom,
    ! [Xa: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Ya)
             => 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,Ya),Z) ) ) ).

% dense_le_bounded
tff(fact_1534_dense__le__bounded,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Ya)
             => 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,Ya),Z) ) ) ).

% dense_le_bounded
tff(fact_1535_dual__order_Oorder__iff__strict,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1536_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
    <=> ( aa_set_int_o(ord_less_set_int(B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1537_dual__order_Oorder__iff__strict,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1538_dual__order_Oorder__iff__strict,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1539_dual__order_Oorder__iff__strict,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1540_dual__order_Oorder__iff__strict,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1541_dual__order_Ostrict__iff__order,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1542_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B2),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1543_dual__order_Ostrict__iff__order,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1544_dual__order_Ostrict__iff__order,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1545_dual__order_Ostrict__iff__order,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1546_dual__order_Ostrict__iff__order,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1547_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1548_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_o(ord_less_set_int(C2),B2)
       => aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1549_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1550_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1551_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1552_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1553_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1554_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(B2),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),B2)
       => aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1555_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1556_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1557_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1558_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( 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),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1559_dual__order_Ostrict__iff__not,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1560_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B2),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B2),A)
        & ~ aa_set_int_o(ord_less_eq_set_int(A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1561_dual__order_Ostrict__iff__not,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1562_dual__order_Ostrict__iff__not,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1563_dual__order_Ostrict__iff__not,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1564_dual__order_Ostrict__iff__not,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1565_order_Ostrict__implies__order,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% order.strict_implies_order
tff(fact_1566_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
     => aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% order.strict_implies_order
tff(fact_1567_order_Ostrict__implies__order,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% order.strict_implies_order
tff(fact_1568_order_Ostrict__implies__order,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ).

% order.strict_implies_order
tff(fact_1569_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% order.strict_implies_order
tff(fact_1570_order_Ostrict__implies__order,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% order.strict_implies_order
tff(fact_1571_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_1572_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B2),A)
     => aa_set_int_o(ord_less_eq_set_int(B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_1573_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_1574_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_1575_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_1576_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_1577_order__le__less,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_less
tff(fact_1578_order__le__less,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
    <=> ( aa_set_int_o(ord_less_set_int(Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_less
tff(fact_1579_order__le__less,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_less
tff(fact_1580_order__le__less,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_less
tff(fact_1581_order__le__less,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_less
tff(fact_1582_order__le__less,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_less
tff(fact_1583_order__less__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
        & ( Xa != Ya ) ) ) ).

% order_less_le
tff(fact_1584_order__less__le,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Ya)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
        & ( Xa != Ya ) ) ) ).

% order_less_le
tff(fact_1585_order__less__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
        & ( Xa != Ya ) ) ) ).

% order_less_le
tff(fact_1586_order__less__le,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
        & ( Xa != Ya ) ) ) ).

% order_less_le
tff(fact_1587_order__less__le,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
        & ( Xa != Ya ) ) ) ).

% order_less_le
tff(fact_1588_order__less__le,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
        & ( Xa != Ya ) ) ) ).

% order_less_le
tff(fact_1589_linorder__not__le,axiom,
    ! [Xa: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ).

% linorder_not_le
tff(fact_1590_linorder__not__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ).

% linorder_not_le
tff(fact_1591_linorder__not__le,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ).

% linorder_not_le
tff(fact_1592_linorder__not__le,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ).

% linorder_not_le
tff(fact_1593_linorder__not__le,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ).

% linorder_not_le
tff(fact_1594_linorder__not__less,axiom,
    ! [Xa: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa) ) ).

% linorder_not_less
tff(fact_1595_linorder__not__less,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa) ) ).

% linorder_not_less
tff(fact_1596_linorder__not__less,axiom,
    ! [Xa: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa) ) ).

% linorder_not_less
tff(fact_1597_linorder__not__less,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa) ) ).

% linorder_not_less
tff(fact_1598_linorder__not__less,axiom,
    ! [Xa: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa) ) ).

% linorder_not_less
tff(fact_1599_order__less__imp__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% order_less_imp_le
tff(fact_1600_order__less__imp__le,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Ya)
     => aa_set_int_o(ord_less_eq_set_int(Xa),Ya) ) ).

% order_less_imp_le
tff(fact_1601_order__less__imp__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ).

% order_less_imp_le
tff(fact_1602_order__less__imp__le,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya) ) ).

% order_less_imp_le
tff(fact_1603_order__less__imp__le,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya) ) ).

% order_less_imp_le
tff(fact_1604_order__less__imp__le,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya) ) ).

% order_less_imp_le
tff(fact_1605_order__le__neq__trans,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ( A != B2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_1606_order__le__neq__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( ( A != B2 )
       => aa_set_int_o(ord_less_set_int(A),B2) ) ) ).

% order_le_neq_trans
tff(fact_1607_order__le__neq__trans,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( A != B2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_1608_order__le__neq__trans,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( A != B2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_1609_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( A != B2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_1610_order__le__neq__trans,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( ( A != B2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_1611_order__neq__le__trans,axiom,
    ! [A: real,B2: real] :
      ( ( A != B2 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_1612_order__neq__le__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A != B2 )
     => ( aa_set_int_o(ord_less_eq_set_int(A),B2)
       => aa_set_int_o(ord_less_set_int(A),B2) ) ) ).

% order_neq_le_trans
tff(fact_1613_order__neq__le__trans,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_1614_order__neq__le__trans,axiom,
    ! [A: num,B2: num] :
      ( ( A != B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_1615_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_1616_order__neq__le__trans,axiom,
    ! [A: int,B2: int] :
      ( ( A != B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_1617_order__le__less__trans,axiom,
    ! [Xa: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_1618_order__le__less__trans,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_set_int(Ya),Z)
       => aa_set_int_o(ord_less_set_int(Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_1619_order__le__less__trans,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),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,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_1620_order__le__less__trans,axiom,
    ! [Xa: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),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,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_1621_order__le__less__trans,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),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,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_1622_order__le__less__trans,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),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,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_1623_order__less__le__trans,axiom,
    ! [Xa: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_1624_order__less__le__trans,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_eq_set_int(Ya),Z)
       => aa_set_int_o(ord_less_set_int(Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_1625_order__less__le__trans,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),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,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_1626_order__less__le__trans,axiom,
    ! [Xa: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),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,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_1627_order__less__le__trans,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),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,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_1628_order__less__le__trans,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),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,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_1629_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1630_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1631_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B2: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1632_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1633_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1634_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B2: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1635_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1636_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1637_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1638_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B2: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_1639_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1640_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1641_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1642_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1643_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1644_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1645_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1646_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1647_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1648_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_1649_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1650_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1651_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1652_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1653_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B2: rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1654_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B2: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1655_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1656_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1657_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B2: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1658_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B2: num,C2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_1659_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1660_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1661_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1662_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B2)),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1663_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_real,C2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B2)),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1664_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B2)),C2)
       => ( ! [X2: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1665_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B2)),C2)
       => ( ! [X2: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1666_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B2)),C2)
       => ( ! [X2: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1667_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B2)),C2)
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1668_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B2)),C2)
       => ( ! [X2: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_1669_linorder__le__less__linear,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ).

% linorder_le_less_linear
tff(fact_1670_linorder__le__less__linear,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ).

% linorder_le_less_linear
tff(fact_1671_linorder__le__less__linear,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xa) ) ).

% linorder_le_less_linear
tff(fact_1672_linorder__le__less__linear,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xa) ) ).

% linorder_le_less_linear
tff(fact_1673_linorder__le__less__linear,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ).

% linorder_le_less_linear
tff(fact_1674_order__le__imp__less__or__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1675_order__le__imp__less__or__eq,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_set_int(Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1676_order__le__imp__less__or__eq,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1677_order__le__imp__less__or__eq,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1678_order__le__imp__less__or__eq,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1679_order__le__imp__less__or__eq,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1680_gr__zeroI,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr_zeroI
tff(fact_1681_not__less__zero,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less_zero
tff(fact_1682_gr__implies__not__zero,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1683_zero__less__iff__neq__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1684_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1685_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_1686_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_1687_bot_Onot__eq__extremum,axiom,
    ! [A: filter_nat] :
      ( ( A != bot_bot_filter_nat )
    <=> ord_less_filter_nat(bot_bot_filter_nat,A) ) ).

% bot.not_eq_extremum
tff(fact_1688_bot_Onot__eq__extremum,axiom,
    ! [A: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
    <=> aa_set_set_nat_o(ord_less_set_set_nat(bot_bot_set_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1689_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),A) ) ).

% bot.not_eq_extremum
tff(fact_1690_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
    <=> aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(bot_bo7653980558646680370d_enat),A) ) ).

% bot.not_eq_extremum
tff(fact_1691_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1692_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1693_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1694_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_1695_bot_Oextremum__strict,axiom,
    ! [A: filter_nat] : ~ ord_less_filter_nat(A,bot_bot_filter_nat) ).

% bot.extremum_strict
tff(fact_1696_bot_Oextremum__strict,axiom,
    ! [A: set_set_nat] : ~ aa_set_set_nat_o(ord_less_set_set_nat(A),bot_bot_set_set_nat) ).

% bot.extremum_strict
tff(fact_1697_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ aa_set_real_o(ord_less_set_real(A),bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_1698_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] : ~ aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A),bot_bo7653980558646680370d_enat) ).

% bot.extremum_strict
tff(fact_1699_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1700_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1701_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_o,B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_set_o(C2),B2)
     => aa_set_o_o(ord_less_set_o(C2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1702_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_int,B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(C2),B2)
     => aa_set_int_o(ord_less_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1703_sup_Ostrict__coboundedI2,axiom,
    ! [C2: filter_nat,B2: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(C2,B2)
     => ord_less_filter_nat(C2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1704_sup_Ostrict__coboundedI2,axiom,
    ! [C2: real,B2: real,A: 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(sup_sup_real,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1705_sup_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1706_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A: 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(sup_sup_nat,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1707_sup_Ostrict__coboundedI2,axiom,
    ! [C2: int,B2: int,A: 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(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_1708_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1709_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_o,A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(C2),A)
     => aa_set_o_o(ord_less_set_o(C2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1710_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(C2),A)
     => aa_set_int_o(ord_less_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1711_sup_Ostrict__coboundedI1,axiom,
    ! [C2: filter_nat,A: filter_nat,B2: filter_nat] :
      ( ord_less_filter_nat(C2,A)
     => ord_less_filter_nat(C2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1712_sup_Ostrict__coboundedI1,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1713_sup_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1714_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1715_sup_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_1716_sup_Ostrict__order__iff,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B2),A)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1717_sup_Ostrict__order__iff,axiom,
    ! [B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_set_o(B2),A)
    <=> ( ( A = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1718_sup_Ostrict__order__iff,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B2),A)
    <=> ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1719_sup_Ostrict__order__iff,axiom,
    ! [B2: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(B2,A)
    <=> ( ( A = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1720_sup_Ostrict__order__iff,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1721_sup_Ostrict__order__iff,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1722_sup_Ostrict__order__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1723_sup_Ostrict__order__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) )
        & ( A != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_1724_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B2),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1725_sup_Ostrict__boundedE,axiom,
    ! [B2: set_o,C2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B2),C2)),A)
     => ~ ( aa_set_o_o(ord_less_set_o(B2),A)
         => ~ aa_set_o_o(ord_less_set_o(C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1726_sup_Ostrict__boundedE,axiom,
    ! [B2: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)),A)
     => ~ ( aa_set_int_o(ord_less_set_int(B2),A)
         => ~ aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1727_sup_Ostrict__boundedE,axiom,
    ! [B2: filter_nat,C2: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C2),A)
     => ~ ( ord_less_filter_nat(B2,A)
         => ~ ord_less_filter_nat(C2,A) ) ) ).

% sup.strict_boundedE
tff(fact_1728_sup_Ostrict__boundedE,axiom,
    ! [B2: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,B2),C2)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1729_sup_Ostrict__boundedE,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B2),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1730_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1731_sup_Ostrict__boundedE,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1732_sup_Oabsorb4,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1733_sup_Oabsorb4,axiom,
    ! [A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(A),B2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1734_sup_Oabsorb4,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1735_sup_Oabsorb4,axiom,
    ! [A: filter_nat,B2: filter_nat] :
      ( ord_less_filter_nat(A,B2)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1736_sup_Oabsorb4,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1737_sup_Oabsorb4,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1738_sup_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1739_sup_Oabsorb4,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_1740_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B2),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1741_sup_Oabsorb3,axiom,
    ! [B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_set_o(B2),A)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1742_sup_Oabsorb3,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B2),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1743_sup_Oabsorb3,axiom,
    ! [B2: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(B2,A)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1744_sup_Oabsorb3,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1745_sup_Oabsorb3,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1746_sup_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1747_sup_Oabsorb3,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2) = A ) ) ).

% sup.absorb3
tff(fact_1748_less__supI2,axiom,
    ! [Xa: set_nat,B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xa),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% less_supI2
tff(fact_1749_less__supI2,axiom,
    ! [Xa: set_o,B2: set_o,A: set_o] :
      ( aa_set_o_o(ord_less_set_o(Xa),B2)
     => aa_set_o_o(ord_less_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% less_supI2
tff(fact_1750_less__supI2,axiom,
    ! [Xa: set_int,B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),B2)
     => aa_set_int_o(ord_less_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% less_supI2
tff(fact_1751_less__supI2,axiom,
    ! [Xa: filter_nat,B2: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(Xa,B2)
     => ord_less_filter_nat(Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% less_supI2
tff(fact_1752_less__supI2,axiom,
    ! [Xa: real,B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B2)) ) ).

% less_supI2
tff(fact_1753_less__supI2,axiom,
    ! [Xa: rat,B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% less_supI2
tff(fact_1754_less__supI2,axiom,
    ! [Xa: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ).

% less_supI2
tff(fact_1755_less__supI2,axiom,
    ! [Xa: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% less_supI2
tff(fact_1756_less__supI1,axiom,
    ! [Xa: set_nat,A: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xa),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2)) ) ).

% less_supI1
tff(fact_1757_less__supI1,axiom,
    ! [Xa: set_o,A: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(Xa),A)
     => aa_set_o_o(ord_less_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B2)) ) ).

% less_supI1
tff(fact_1758_less__supI1,axiom,
    ! [Xa: set_int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),A)
     => aa_set_int_o(ord_less_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B2)) ) ).

% less_supI1
tff(fact_1759_less__supI1,axiom,
    ! [Xa: filter_nat,A: filter_nat,B2: filter_nat] :
      ( ord_less_filter_nat(Xa,A)
     => ord_less_filter_nat(Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A),B2)) ) ).

% less_supI1
tff(fact_1760_less__supI1,axiom,
    ! [Xa: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B2)) ) ).

% less_supI1
tff(fact_1761_less__supI1,axiom,
    ! [Xa: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B2)) ) ).

% less_supI1
tff(fact_1762_less__supI1,axiom,
    ! [Xa: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B2)) ) ).

% less_supI1
tff(fact_1763_less__supI1,axiom,
    ! [Xa: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B2)) ) ).

% less_supI1
tff(fact_1764_arg__min__if__finite_I2_J,axiom,
    ! [S: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ( S != bot_bot_set_complex )
       => ~ ? [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),S)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(F,lattic8794016678065449205x_real(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1765_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),S)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,lattic488527866317076247t_real(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1766_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( aa_set_int_o(member_int(X4),S)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,lattic2675449441010098035t_real(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1767_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(X4))),aa_o_real(F,aa_set_o_o(lattic8697145971487455083o_real(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1768_arg__min__if__finite_I2_J,axiom,
    ! [S: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ( S != bot_bot_set_real )
       => ~ ? [X4: real] :
              ( aa_set_real_o(member_real(X4),S)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,lattic8440615504127631091l_real(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1769_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ~ ? [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,lattic1189837152898106425t_real(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1770_arg__min__if__finite_I2_J,axiom,
    ! [S: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ( S != bot_bot_set_complex )
       => ~ ? [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),S)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1771_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),S)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,lattic6811802900495863747at_rat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1772_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( aa_set_int_o(member_int(X4),S)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1773_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(X4))),aa_o_rat(F,aa_set_o_o(lattic2140725968369957399_o_rat(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1774_infinite__growing,axiom,
    ! [X3: set_o] :
      ( ( X3 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),X3)
           => ? [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),X3)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X2)),(Xa2)) ) )
       => ~ aa_set_o_o(finite_finite_o,X3) ) ) ).

% infinite_growing
tff(fact_1775_infinite__growing,axiom,
    ! [X3: set_Extended_enat] :
      ( ( X3 != bot_bo7653980558646680370d_enat )
     => ( ! [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),X3)
           => ? [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),X3)
                & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X2),Xa2) ) )
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,X3) ) ) ).

% infinite_growing
tff(fact_1776_infinite__growing,axiom,
    ! [X3: set_real] :
      ( ( X3 != bot_bot_set_real )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),X3)
           => ? [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),X3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Xa2) ) )
       => ~ aa_set_real_o(finite_finite_real,X3) ) ) ).

% infinite_growing
tff(fact_1777_infinite__growing,axiom,
    ! [X3: set_rat] :
      ( ( X3 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),X3)
           => ? [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),X3)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Xa2) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X3) ) ) ).

% infinite_growing
tff(fact_1778_infinite__growing,axiom,
    ! [X3: set_num] :
      ( ( X3 != bot_bot_set_num )
     => ( ! [X2: num] :
            ( aa_set_num_o(member_num(X2),X3)
           => ? [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),X3)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Xa2) ) )
       => ~ aa_set_num_o(finite_finite_num,X3) ) ) ).

% infinite_growing
tff(fact_1779_infinite__growing,axiom,
    ! [X3: set_nat] :
      ( ( X3 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),X3)
           => ? [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),X3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa2) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X3) ) ) ).

% infinite_growing
tff(fact_1780_infinite__growing,axiom,
    ! [X3: set_int] :
      ( ( X3 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),X3)
           => ? [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),X3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa2) ) )
       => ~ aa_set_int_o(finite_finite_int,X3) ) ) ).

% infinite_growing
tff(fact_1781_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),S)
            & ~ ? [Xa2: $o] :
                  ( aa_set_o_o(member_o((Xa2)),S)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa2)),(X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1782_ex__min__if__finite,axiom,
    ! [S: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( ( S != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),S)
            & ~ ? [Xa2: set_nat] :
                  ( aa_set_set_nat_o(member_set_nat(Xa2),S)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1783_ex__min__if__finite,axiom,
    ! [S: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ? [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),S)
            & ~ ? [Xa2: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),S)
                  & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1784_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ( S != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),S)
            & ~ ? [Xa2: real] :
                  ( aa_set_real_o(member_real(Xa2),S)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1785_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S)
     => ( ( S != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),S)
            & ~ ? [Xa2: rat] :
                  ( aa_set_rat_o(member_rat(Xa2),S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1786_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( aa_set_num_o(finite_finite_num,S)
     => ( ( S != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),S)
            & ~ ? [Xa2: num] :
                  ( aa_set_num_o(member_num(Xa2),S)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1787_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),S)
            & ~ ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1788_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),S)
            & ~ ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1789_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),lattic8928443293348198069n_real(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1790_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),lattic3835124923745554447et_nat(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1791_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((A)),A2)
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1792_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),lattic5005175426920976669d_enat(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1793_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => aa_set_int_o(ord_less_eq_set_int(A),lattic8880645941091133547et_int(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1794_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),lattic458866745392299617in_rat(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1795_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),lattic1093996805478795353in_nat(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1796_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),lattic1091506334969745077in_int(A2)) ) ) ).

% Sup_fin.coboundedI
tff(fact_1797_Sup__fin_Oin__idem,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,Xa),lattic8928443293348198069n_real(A2)) = lattic8928443293348198069n_real(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1798_Sup__fin_Oin__idem,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( aa_o_o(aa_o_fun_o_o(sup_sup_o,(Xa)),aa_set_o_o(lattic1508158080041050831_fin_o,A2))
        <=> aa_set_o_o(lattic1508158080041050831_fin_o,A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1799_Sup__fin_Oin__idem,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),lattic1091506334969745077in_int(A2)) = lattic1091506334969745077in_int(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1800_Sup__fin_Oin__idem,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( sup_su3973961784419623482d_enat(Xa,lattic5005175426920976669d_enat(A2)) = lattic5005175426920976669d_enat(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1801_Sup__fin_Oin__idem,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),lattic3835124923745554447et_nat(A2)) = lattic3835124923745554447et_nat(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1802_Sup__fin_Oin__idem,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),lattic1093996805478795353in_nat(A2)) = lattic1093996805478795353in_nat(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1803_Sup__fin_Oin__idem,axiom,
    ! [A2: set_set_o,Xa: set_o] :
      ( finite_finite_set_o(A2)
     => ( member_set_o(Xa,A2)
       => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),lattic3158155371183623599_set_o(A2)) = lattic3158155371183623599_set_o(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1804_Sup__fin_Oin__idem,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(Xa,A2)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),lattic8880645941091133547et_int(A2)) = lattic8880645941091133547et_int(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1805_Sup__fin_Oin__idem,axiom,
    ! [A2: set_filter_nat,Xa: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( member_filter_nat(Xa,A2)
       => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),lattic5930898082463196905er_nat(A2)) = lattic5930898082463196905er_nat(A2) ) ) ) ).

% Sup_fin.in_idem
tff(fact_1806_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B4: $o,A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A5)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B4)),(X4)) )
               => ( aa_set_o_o(P,A5)
                 => aa_set_o_o(P,insert_o((B4),A5)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1807_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [B4: extended_enat,A5: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
             => ( ! [X4: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A5)
                   => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B4),X4) )
               => ( aa_set3337841544853333262enat_o(P,A5)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(B4,A5)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1808_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B4: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X4: real] :
                    ( aa_set_real_o(member_real(X4),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B4),X4) )
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,insert_real(B4,A5)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1809_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B4: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),X4) )
               => ( aa_set_rat_o(P,A5)
                 => aa_set_rat_o(P,insert_rat(B4,A5)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1810_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B4: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X4: num] :
                    ( aa_set_num_o(member_num(X4),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B4),X4) )
               => ( aa_set_num_o(P,A5)
                 => aa_set_num_o(P,insert_num(B4,A5)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1811_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B4: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B4),X4) )
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,insert_nat(B4,A5)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1812_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B4: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B4),X4) )
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,insert_int(B4,A5)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1813_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B4: $o,A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A5)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X4)),(B4)) )
               => ( aa_set_o_o(P,A5)
                 => aa_set_o_o(P,insert_o((B4),A5)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1814_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [B4: extended_enat,A5: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
             => ( ! [X4: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A5)
                   => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),B4) )
               => ( aa_set3337841544853333262enat_o(P,A5)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(B4,A5)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1815_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B4: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X4: real] :
                    ( aa_set_real_o(member_real(X4),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B4) )
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,insert_real(B4,A5)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1816_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B4: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),B4) )
               => ( aa_set_rat_o(P,A5)
                 => aa_set_rat_o(P,insert_rat(B4,A5)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1817_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B4: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X4: num] :
                    ( aa_set_num_o(member_num(X4),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X4),B4) )
               => ( aa_set_num_o(P,A5)
                 => aa_set_num_o(P,insert_num(B4,A5)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1818_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B4: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),B4) )
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,insert_nat(B4,A5)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1819_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B4: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),B4) )
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,insert_int(B4,A5)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1820_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1821_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1822_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1823_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1824_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_1825_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),(Xa))
        <=> ! [X: $o] :
              ( aa_set_o_o(member_o((X)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(Xa)) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1826_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),Xa)
        <=> ! [X: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(X),A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1827_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic8928443293348198069n_real(A2)),Xa)
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1828_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic5005175426920976669d_enat(A2)),Xa)
        <=> ! [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1829_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(lattic8880645941091133547et_int(A2)),Xa)
        <=> ! [X: set_int] :
              ( member_set_int(X,A2)
             => aa_set_int_o(ord_less_eq_set_int(X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1830_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A2)),Xa)
        <=> ! [X: rat] :
              ( aa_set_rat_o(member_rat(X),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1831_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),Xa)
        <=> ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1832_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),Xa)
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_1833_Sup__fin_OboundedI,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(Xa)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),(Xa)) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1834_Sup__fin_OboundedI,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(A3),A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),Xa) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1835_Sup__fin_OboundedI,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),Xa) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic8928443293348198069n_real(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1836_Sup__fin_OboundedI,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [A3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A3),Xa) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic5005175426920976669d_enat(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1837_Sup__fin_OboundedI,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( ! [A3: set_int] :
              ( member_set_int(A3,A2)
             => aa_set_int_o(ord_less_eq_set_int(A3),Xa) )
         => aa_set_int_o(ord_less_eq_set_int(lattic8880645941091133547et_int(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1838_Sup__fin_OboundedI,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),Xa) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1839_Sup__fin_OboundedI,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),Xa) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1840_Sup__fin_OboundedI,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),Xa) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),Xa) ) ) ) ).

% Sup_fin.boundedI
tff(fact_1841_Sup__fin_OboundedE,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),(Xa))
         => ! [A8: $o] :
              ( aa_set_o_o(member_o((A8)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A8)),(Xa)) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1842_Sup__fin_OboundedE,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),Xa)
         => ! [A8: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(A8),A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1843_Sup__fin_OboundedE,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic8928443293348198069n_real(A2)),Xa)
         => ! [A8: real] :
              ( aa_set_real_o(member_real(A8),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1844_Sup__fin_OboundedE,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic5005175426920976669d_enat(A2)),Xa)
         => ! [A8: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A8),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1845_Sup__fin_OboundedE,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(lattic8880645941091133547et_int(A2)),Xa)
         => ! [A8: set_int] :
              ( member_set_int(A8,A2)
             => aa_set_int_o(ord_less_eq_set_int(A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1846_Sup__fin_OboundedE,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A2)),Xa)
         => ! [A8: rat] :
              ( aa_set_rat_o(member_rat(A8),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1847_Sup__fin_OboundedE,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),Xa)
         => ! [A8: nat] :
              ( aa_set_nat_o(member_nat(A8),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1848_Sup__fin_OboundedE,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),Xa)
         => ! [A8: int] :
              ( aa_set_int_o(member_int(A8),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A8),Xa) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_1849_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_set_nat_o(member_nat(N),S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1850_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),aa_set_o_o(lattic1508158080041050831_fin_o,B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1851_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),lattic3835124923745554447et_nat(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1852_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_real_o(finite_finite_real,B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic8928443293348198069n_real(A2)),lattic8928443293348198069n_real(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1853_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic5005175426920976669d_enat(A2)),lattic5005175426920976669d_enat(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1854_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_set_int,B: set_set_int] :
      ( ord_le4403425263959731960et_int(A2,B)
     => ( ( A2 != bot_bot_set_set_int )
       => ( finite6197958912794628473et_int(B)
         => aa_set_int_o(ord_less_eq_set_int(lattic8880645941091133547et_int(A2)),lattic8880645941091133547et_int(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1855_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_rat,B: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A2),B)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A2)),lattic458866745392299617in_rat(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1856_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),lattic1093996805478795353in_nat(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1857_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),lattic1091506334969745077in_int(B)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_1858_Sup__fin_Osubset,axiom,
    ! [A2: set_set_o,B: set_set_o] :
      ( finite_finite_set_o(A2)
     => ( ( B != bot_bot_set_set_o )
       => ( ord_le4374716579403074808_set_o(B,A2)
         => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,lattic3158155371183623599_set_o(B)),lattic3158155371183623599_set_o(A2)) = lattic3158155371183623599_set_o(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1859_Sup__fin_Osubset,axiom,
    ! [A2: set_set_int,B: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( B != bot_bot_set_set_int )
       => ( ord_le4403425263959731960et_int(B,A2)
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,lattic8880645941091133547et_int(B)),lattic8880645941091133547et_int(A2)) = lattic8880645941091133547et_int(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1860_Sup__fin_Osubset,axiom,
    ! [A2: set_filter_nat,B: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( B != bot_bo498966703094740906er_nat )
       => ( ord_le2426478655948331894er_nat(B,A2)
         => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,lattic5930898082463196905er_nat(B)),lattic5930898082463196905er_nat(A2)) = lattic5930898082463196905er_nat(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1861_Sup__fin_Osubset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( B != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic1093996805478795353in_nat(B)),lattic1093996805478795353in_nat(A2)) = lattic1093996805478795353in_nat(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1862_Sup__fin_Osubset,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( B != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B),A2)
         => ( aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,B)),aa_set_o_o(lattic1508158080041050831_fin_o,A2))
          <=> aa_set_o_o(lattic1508158080041050831_fin_o,A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1863_Sup__fin_Osubset,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( B != bot_bot_set_set_nat )
       => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B),A2)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(B)),lattic3835124923745554447et_nat(A2)) = lattic3835124923745554447et_nat(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1864_Sup__fin_Osubset,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( B != bot_bot_set_real )
       => ( aa_set_real_o(ord_less_eq_set_real(B),A2)
         => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,lattic8928443293348198069n_real(B)),lattic8928443293348198069n_real(A2)) = lattic8928443293348198069n_real(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1865_Sup__fin_Osubset,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( B != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B),A2)
         => ( sup_su3973961784419623482d_enat(lattic5005175426920976669d_enat(B),lattic5005175426920976669d_enat(A2)) = lattic5005175426920976669d_enat(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1866_Sup__fin_Osubset,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( B != bot_bot_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(B),A2)
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic1091506334969745077in_int(B)),lattic1091506334969745077in_int(A2)) = lattic1091506334969745077in_int(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1867_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_set_o,Xa: set_o] :
      ( finite_finite_set_o(A2)
     => ( ~ member_set_o(Xa,A2)
       => ( ( A2 != bot_bot_set_set_o )
         => ( lattic3158155371183623599_set_o(insert_set_o(Xa,A2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),lattic3158155371183623599_set_o(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1868_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ~ member_set_int(Xa,A2)
       => ( ( A2 != bot_bot_set_set_int )
         => ( lattic8880645941091133547et_int(insert_set_int(Xa,A2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),lattic8880645941091133547et_int(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1869_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_filter_nat,Xa: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ~ member_filter_nat(Xa,A2)
       => ( ( A2 != bot_bo498966703094740906er_nat )
         => ( lattic5930898082463196905er_nat(insert_filter_nat(Xa,A2)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),lattic5930898082463196905er_nat(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1870_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( lattic1093996805478795353in_nat(insert_nat(Xa,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),lattic1093996805478795353in_nat(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1871_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xa),A2)
       => ( ( A2 != bot_bot_set_int )
         => ( lattic1091506334969745077in_int(insert_int(Xa,A2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),lattic1091506334969745077in_int(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1872_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((Xa)),A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((Xa),A2))
          <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(Xa)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1873_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( ( A2 != bot_bot_set_set_nat )
         => ( lattic3835124923745554447et_nat(insert_set_nat(Xa,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),lattic3835124923745554447et_nat(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1874_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xa),A2)
       => ( ( A2 != bot_bot_set_real )
         => ( lattic8928443293348198069n_real(insert_real(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,Xa),lattic8928443293348198069n_real(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1875_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( lattic5005175426920976669d_enat(insert_Extended_enat(Xa,A2)) = sup_su3973961784419623482d_enat(Xa,lattic5005175426920976669d_enat(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1876_Sup__fin_Oclosed,axiom,
    ! [A2: set_set_o] :
      ( finite_finite_set_o(A2)
     => ( ( A2 != bot_bot_set_set_o )
       => ( ! [X2: set_o,Y: set_o] : member_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X2),Y),insert_set_o(X2,insert_set_o(Y,bot_bot_set_set_o)))
         => member_set_o(lattic3158155371183623599_set_o(A2),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1877_Sup__fin_Oclosed,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( ! [X2: set_int,Y: set_int] : member_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X2),Y),insert_set_int(X2,insert_set_int(Y,bot_bot_set_set_int)))
         => member_set_int(lattic8880645941091133547et_int(A2),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1878_Sup__fin_Oclosed,axiom,
    ! [A2: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ( ! [X2: filter_nat,Y: filter_nat] : member_filter_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X2),Y),insert_filter_nat(X2,insert_filter_nat(Y,bot_bo498966703094740906er_nat)))
         => member_filter_nat(lattic5930898082463196905er_nat(A2),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1879_Sup__fin_Oclosed,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat,Y: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y)),insert_nat(X2,insert_nat(Y,bot_bot_set_nat)))
         => aa_set_nat_o(member_nat(lattic1093996805478795353in_nat(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1880_Sup__fin_Oclosed,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int,Y: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(sup_sup_int,X2),Y)),insert_int(X2,insert_int(Y,bot_bot_set_int)))
         => aa_set_int_o(member_int(lattic1091506334969745077in_int(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1881_Sup__fin_Oclosed,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o,Y: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(sup_sup_o,(X2)),(Y))),insert_o((X2),insert_o((Y),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1508158080041050831_fin_o,A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1882_Sup__fin_Oclosed,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y: set_nat] : aa_set_set_nat_o(member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y)),insert_set_nat(X2,insert_set_nat(Y,bot_bot_set_set_nat)))
         => aa_set_set_nat_o(member_set_nat(lattic3835124923745554447et_nat(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1883_Sup__fin_Oclosed,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real,Y: real] : aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,X2),Y)),insert_real(X2,insert_real(Y,bot_bot_set_real)))
         => aa_set_real_o(member_real(lattic8928443293348198069n_real(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1884_Sup__fin_Oclosed,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X2: extended_enat,Y: extended_enat] : aa_set3337841544853333262enat_o(member_Extended_enat(sup_su3973961784419623482d_enat(X2,Y)),insert_Extended_enat(X2,insert_Extended_enat(Y,bot_bo7653980558646680370d_enat)))
         => aa_set3337841544853333262enat_o(member_Extended_enat(lattic5005175426920976669d_enat(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1885_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_1886_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_1887_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1888_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1889_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1890_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1891_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A7: $o,B6: $o] : Ta = vEBT_Leaf((A7),(B6)) ) ).

% deg1Leaf
tff(fact_1892_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A3: $o,B4: $o] : Ta = vEBT_Leaf((A3),(B4)) ) ).

% deg_1_Leaf
tff(fact_1893_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( Na = one_one_nat )
       => ? [A3: $o,B4: $o] : Ta = vEBT_Leaf((A3),(B4)) ) ) ).

% deg_1_Leafy
tff(fact_1894_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1895_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,Na: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Na) )
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% bounded_nat_set_is_finite
tff(fact_1896_psubsetI,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( ( A2 != B )
       => aa_set_int_o(ord_less_set_int(A2),B) ) ) ).

% psubsetI
tff(fact_1897_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_1898_one__reorient,axiom,
    ! [Xa: complex] :
      ( ( one_one_complex = Xa )
    <=> ( Xa = one_one_complex ) ) ).

% one_reorient
tff(fact_1899_one__reorient,axiom,
    ! [Xa: real] :
      ( ( one_one_real = Xa )
    <=> ( Xa = one_one_real ) ) ).

% one_reorient
tff(fact_1900_one__reorient,axiom,
    ! [Xa: rat] :
      ( ( one_one_rat = Xa )
    <=> ( Xa = one_one_rat ) ) ).

% one_reorient
tff(fact_1901_one__reorient,axiom,
    ! [Xa: nat] :
      ( ( one_one_nat = Xa )
    <=> ( Xa = one_one_nat ) ) ).

% one_reorient
tff(fact_1902_one__reorient,axiom,
    ! [Xa: int] :
      ( ( one_one_int = Xa )
    <=> ( Xa = one_one_int ) ) ).

% one_reorient
tff(fact_1903_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1904_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(ord_less_set_int(A2),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1905_not__psubset__empty,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(ord_less_set_o(A2),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1906_not__psubset__empty,axiom,
    ! [A2: set_set_nat] : ~ aa_set_set_nat_o(ord_less_set_set_nat(A2),bot_bot_set_set_nat) ).

% not_psubset_empty
tff(fact_1907_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ aa_set_real_o(ord_less_set_real(A2),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_1908_not__psubset__empty,axiom,
    ! [A2: set_Extended_enat] : ~ aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),bot_bo7653980558646680370d_enat) ).

% not_psubset_empty
tff(fact_1909_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [A5: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A5)
           => ( ! [B7: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B7),A5)
                 => aa_set_nat_o(P,B7) )
             => aa_set_nat_o(P,A5) ) )
       => aa_set_nat_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1910_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [A5: set_int] :
            ( aa_set_int_o(finite_finite_int,A5)
           => ( ! [B7: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B7),A5)
                 => aa_set_int_o(P,B7) )
             => aa_set_int_o(P,A5) ) )
       => aa_set_int_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1911_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [A5: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A5)
           => ( ! [B7: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B7),A5)
                 => aa_set_complex_o(P,B7) )
             => aa_set_complex_o(P,A5) ) )
       => aa_set_complex_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1912_finite__psubset__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [A5: set_Extended_enat] :
            ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
           => ( ! [B7: set_Extended_enat] :
                  ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(B7),A5)
                 => aa_set3337841544853333262enat_o(P,B7) )
             => aa_set3337841544853333262enat_o(P,A5) ) )
       => aa_set3337841544853333262enat_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1913_psubsetE,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A2),B)
         => aa_set_int_o(ord_less_eq_set_int(B),A2) ) ) ).

% psubsetE
tff(fact_1914_psubset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B)
        & ( A2 != B ) ) ) ).

% psubset_eq
tff(fact_1915_psubset__imp__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
     => aa_set_int_o(ord_less_eq_set_int(A2),B) ) ).

% psubset_imp_subset
tff(fact_1916_psubset__subset__trans,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_set_int(A2),C) ) ) ).

% psubset_subset_trans
tff(fact_1917_subset__not__subset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B)
        & ~ aa_set_int_o(ord_less_eq_set_int(B),A2) ) ) ).

% subset_not_subset_eq
tff(fact_1918_subset__psubset__trans,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(ord_less_set_int(B),C)
       => aa_set_int_o(ord_less_set_int(A2),C) ) ) ).

% subset_psubset_trans
tff(fact_1919_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
    <=> ( aa_set_int_o(ord_less_set_int(A2),B)
        | ( A2 = B ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1929_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_1930_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_1931_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_1932_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_1933_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B2))),Xa)
    <=> $ite(
          Xa = zero_zero_nat,
          (A),
          $ite(Xa = one_one_nat,(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_1934_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B2))),Xa)
    <=> $ite(
          Xa = zero_zero_nat,
          (A),
          $ite(Xa = one_one_nat,(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1935_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xa: nat,M5: nat] :
      ( aa_nat_o(P,Xa)
     => ( ! [X2: nat] :
            ( aa_nat_o(P,X2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M5) )
       => ~ ! [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_1936_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),M) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1937_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Xa: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B2)),Xa) = $ite(
        Xa = zero_zero_nat,
        vEBT_Leaf($true,(B2)),
        $ite(Xa = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_nat__descend__induct,axiom,
    ! [Na: nat,P: fun_nat_o,M2: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K2)
         => aa_nat_o(P,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
           => ( ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I3)
                 => aa_nat_o(P,I3) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,M2) ) ) ).

% nat_descend_induct
tff(fact_1959_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1960_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1961_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1962_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1963_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1964_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)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1965_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)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1966_complete__interval,axiom,
    ! [A: real,B2: real,P: fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(P,A)
       => ( ~ aa_real_o(P,B2)
         => ? [C4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C4),B2)
              & ! [X4: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),C4) )
                 => aa_real_o(P,X4) )
              & ! [D3: real] :
                  ( ! [X2: real] :
                      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                        & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),D3) )
                     => aa_real_o(P,X2) )
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D3),C4) ) ) ) ) ) ).

% complete_interval
tff(fact_1967_complete__interval,axiom,
    ! [A: nat,B2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(P,A)
       => ( ~ aa_nat_o(P,B2)
         => ? [C4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C4),B2)
              & ! [X4: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),C4) )
                 => aa_nat_o(P,X4) )
              & ! [D3: nat] :
                  ( ! [X2: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X2)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),D3) )
                     => aa_nat_o(P,X2) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C4) ) ) ) ) ) ).

% complete_interval
tff(fact_1968_complete__interval,axiom,
    ! [A: int,B2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(P,A)
       => ( ~ aa_int_o(P,B2)
         => ? [C4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C4)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C4),B2)
              & ! [X4: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X4)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X4),C4) )
                 => aa_int_o(P,X4) )
              & ! [D3: int] :
                  ( ! [X2: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X2)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),D3) )
                     => aa_int_o(P,X2) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C4) ) ) ) ) ) ).

% complete_interval
tff(fact_1969_verit__comp__simplify1_I3_J,axiom,
    ! [B8: real,A9: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B8),A9)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A9),B8) ) ).

% verit_comp_simplify1(3)
tff(fact_1970_verit__comp__simplify1_I3_J,axiom,
    ! [B8: rat,A9: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B8),A9)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A9),B8) ) ).

% verit_comp_simplify1(3)
tff(fact_1971_verit__comp__simplify1_I3_J,axiom,
    ! [B8: num,A9: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B8),A9)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A9),B8) ) ).

% verit_comp_simplify1(3)
tff(fact_1972_verit__comp__simplify1_I3_J,axiom,
    ! [B8: nat,A9: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B8),A9)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A9),B8) ) ).

% verit_comp_simplify1(3)
tff(fact_1973_verit__comp__simplify1_I3_J,axiom,
    ! [B8: int,A9: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B8),A9)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A9),B8) ) ).

% verit_comp_simplify1(3)
tff(fact_1974_psubsetD,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,C2: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),B)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
       => aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% psubsetD
tff(fact_1975_psubsetD,axiom,
    ! [A2: set_real,B: set_real,C2: real] :
      ( aa_set_real_o(ord_less_set_real(A2),B)
     => ( aa_set_real_o(member_real(C2),A2)
       => aa_set_real_o(member_real(C2),B) ) ) ).

% psubsetD
tff(fact_1976_psubsetD,axiom,
    ! [A2: set_set_nat,B: set_set_nat,C2: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B)
     => ( aa_set_set_nat_o(member_set_nat(C2),A2)
       => aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% psubsetD
tff(fact_1977_psubsetD,axiom,
    ! [A2: set_nat,B: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
     => ( aa_set_nat_o(member_nat(C2),A2)
       => aa_set_nat_o(member_nat(C2),B) ) ) ).

% psubsetD
tff(fact_1978_psubsetD,axiom,
    ! [A2: set_int,B: set_int,C2: int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
     => ( aa_set_int_o(member_int(C2),A2)
       => aa_set_int_o(member_int(C2),B) ) ) ).

% psubsetD
tff(fact_1979_psubsetD,axiom,
    ! [A2: set_o,B: set_o,C2: $o] :
      ( aa_set_o_o(ord_less_set_o(A2),B)
     => ( aa_set_o_o(member_o((C2)),A2)
       => aa_set_o_o(member_o((C2)),B) ) ) ).

% psubsetD
tff(fact_1980_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1981_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1982_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1983_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1984_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1985_verit__la__disequality,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% verit_la_disequality
tff(fact_1986_verit__la__disequality,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ).

% verit_la_disequality
tff(fact_1987_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ).

% verit_la_disequality
tff(fact_1988_verit__la__disequality,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% verit_la_disequality
tff(fact_1989_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1990_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1991_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1992_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1993_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1994_linorder__neqE__linordered__idom,axiom,
    ! [Xa: real,Ya: real] :
      ( ( Xa != Ya )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1995_linorder__neqE__linordered__idom,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( Xa != Ya )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1996_linorder__neqE__linordered__idom,axiom,
    ! [Xa: int,Ya: int] :
      ( ( Xa != Ya )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1997_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B4)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,B4),A) ) ).

% ex_gt_or_lt
tff(fact_1998_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_1999_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_2000_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_2001_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_2002_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_2003_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_2004_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_2005_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_2006_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_2007_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_2008_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_2009_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_2010_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_2011_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_2012_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_2013_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_2014_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_2015_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_2016_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_2017_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_2018_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_2019_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_2020_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_2021_dbl__inc__simps_I2_J,axiom,
    neg_nu5851722552734809277nc_int(zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_2022_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_2023_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_set_o,Xa: set_o] :
      ( finite_finite_set_o(A2)
     => ( lattic3158155371183623599_set_o(insert_set_o(Xa,A2)) = $ite(minus_4899875422681990719_set_o(A2,insert_set_o(Xa,bot_bot_set_set_o)) = bot_bot_set_set_o,Xa,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),lattic3158155371183623599_set_o(minus_4899875422681990719_set_o(A2,insert_set_o(Xa,bot_bot_set_set_o))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2024_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( lattic8880645941091133547et_int(insert_set_int(Xa,A2)) = $ite(minus_8897228262479074673et_int(A2,insert_set_int(Xa,bot_bot_set_set_int)) = bot_bot_set_set_int,Xa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),lattic8880645941091133547et_int(minus_8897228262479074673et_int(A2,insert_set_int(Xa,bot_bot_set_set_int))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2025_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_filter_nat,Xa: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( lattic5930898082463196905er_nat(insert_filter_nat(Xa,A2)) = $ite(minus_1488485472792771695er_nat(A2,insert_filter_nat(Xa,bot_bo498966703094740906er_nat)) = bot_bo498966703094740906er_nat,Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),lattic5930898082463196905er_nat(minus_1488485472792771695er_nat(A2,insert_filter_nat(Xa,bot_bo498966703094740906er_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2026_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( lattic1091506334969745077in_int(insert_int(Xa,A2)) = $ite(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)) = bot_bot_set_int,Xa,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),lattic1091506334969745077in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2027_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((Xa),A2))
      <=> $ite(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)) = bot_bot_set_o,(Xa),aa_o_o(aa_o_fun_o_o(sup_sup_o,(Xa)),aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2028_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( lattic3835124923745554447et_nat(insert_set_nat(Xa,A2)) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)) = bot_bot_set_set_nat,Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2029_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( lattic8928443293348198069n_real(insert_real(Xa,A2)) = $ite(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)) = bot_bot_set_real,Xa,aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,Xa),lattic8928443293348198069n_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2030_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( lattic5005175426920976669d_enat(insert_Extended_enat(Xa,A2)) = $ite(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = bot_bo7653980558646680370d_enat,Xa,sup_su3973961784419623482d_enat(Xa,lattic5005175426920976669d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2031_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( lattic1093996805478795353in_nat(insert_nat(Xa,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)) = bot_bot_set_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2032_Sup__fin_Oremove,axiom,
    ! [A2: set_set_o,Xa: set_o] :
      ( finite_finite_set_o(A2)
     => ( member_set_o(Xa,A2)
       => ( lattic3158155371183623599_set_o(A2) = $ite(minus_4899875422681990719_set_o(A2,insert_set_o(Xa,bot_bot_set_set_o)) = bot_bot_set_set_o,Xa,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),lattic3158155371183623599_set_o(minus_4899875422681990719_set_o(A2,insert_set_o(Xa,bot_bot_set_set_o))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2033_Sup__fin_Oremove,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(Xa,A2)
       => ( lattic8880645941091133547et_int(A2) = $ite(minus_8897228262479074673et_int(A2,insert_set_int(Xa,bot_bot_set_set_int)) = bot_bot_set_set_int,Xa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),lattic8880645941091133547et_int(minus_8897228262479074673et_int(A2,insert_set_int(Xa,bot_bot_set_set_int))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2034_Sup__fin_Oremove,axiom,
    ! [A2: set_filter_nat,Xa: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( member_filter_nat(Xa,A2)
       => ( lattic5930898082463196905er_nat(A2) = $ite(minus_1488485472792771695er_nat(A2,insert_filter_nat(Xa,bot_bo498966703094740906er_nat)) = bot_bo498966703094740906er_nat,Xa,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),lattic5930898082463196905er_nat(minus_1488485472792771695er_nat(A2,insert_filter_nat(Xa,bot_bo498966703094740906er_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2035_Sup__fin_Oremove,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( lattic1091506334969745077in_int(A2) = $ite(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)) = bot_bot_set_int,Xa,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),lattic1091506334969745077in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2036_Sup__fin_Oremove,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,A2)
        <=> $ite(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)) = bot_bot_set_o,(Xa),aa_o_o(aa_o_fun_o_o(sup_sup_o,(Xa)),aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2037_Sup__fin_Oremove,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( lattic3835124923745554447et_nat(A2) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)) = bot_bot_set_set_nat,Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2038_Sup__fin_Oremove,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( lattic8928443293348198069n_real(A2) = $ite(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)) = bot_bot_set_real,Xa,aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,Xa),lattic8928443293348198069n_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2039_Sup__fin_Oremove,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( lattic5005175426920976669d_enat(A2) = $ite(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = bot_bo7653980558646680370d_enat,Xa,sup_su3973961784419623482d_enat(Xa,lattic5005175426920976669d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2040_Sup__fin_Oremove,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( lattic1093996805478795353in_nat(A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)) = bot_bot_set_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2041_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_2042_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,A2)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2043_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(A2)),lattic3835124923745554447et_nat(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2044_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic2677971596711400399n_real(A2)),lattic8928443293348198069n_real(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2045_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic974744108425517955d_enat(A2)),lattic5005175426920976669d_enat(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2046_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => aa_set_int_o(ord_less_eq_set_int(lattic8060154151401097861et_int(A2)),lattic8880645941091133547et_int(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2047_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic4603258475043424379in_rat(A2)),lattic458866745392299617in_rat(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2048_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic5238388535129920115in_nat(A2)),lattic1093996805478795353in_nat(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2049_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(A2)),lattic1091506334969745077in_int(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2050_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2051_DiffI,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),B)
       => aa_set3337841544853333262enat_o(member_Extended_enat(C2),minus_925952699566721837d_enat(A2,B)) ) ) ).

% DiffI
tff(fact_2052_DiffI,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),A2)
     => ( ~ aa_set_real_o(member_real(C2),B)
       => aa_set_real_o(member_real(C2),minus_minus_set_real(A2,B)) ) ) ).

% DiffI
tff(fact_2053_DiffI,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(C2),B)
       => aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A2,B)) ) ) ).

% DiffI
tff(fact_2054_DiffI,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),A2)
     => ( ~ aa_set_int_o(member_int(C2),B)
       => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) ) ) ).

% DiffI
tff(fact_2055_DiffI,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),A2)
     => ( ~ aa_set_o_o(member_o((C2)),B)
       => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)) ) ) ).

% DiffI
tff(fact_2056_DiffI,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A2)
     => ( ~ aa_set_nat_o(member_nat(C2),B)
       => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) ) ) ).

% DiffI
tff(fact_2057_Diff__iff,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),minus_925952699566721837d_enat(A2,B))
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
        & ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% Diff_iff
tff(fact_2058_Diff__iff,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A2,B))
    <=> ( aa_set_real_o(member_real(C2),A2)
        & ~ aa_set_real_o(member_real(C2),B) ) ) ).

% Diff_iff
tff(fact_2059_Diff__iff,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A2,B))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A2)
        & ~ aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% Diff_iff
tff(fact_2060_Diff__iff,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))
    <=> ( aa_set_int_o(member_int(C2),A2)
        & ~ aa_set_int_o(member_int(C2),B) ) ) ).

% Diff_iff
tff(fact_2061_Diff__iff,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B))
    <=> ( aa_set_o_o(member_o((C2)),A2)
        & ~ aa_set_o_o(member_o((C2)),B) ) ) ).

% Diff_iff
tff(fact_2062_Diff__iff,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))
    <=> ( aa_set_nat_o(member_nat(C2),A2)
        & ~ aa_set_nat_o(member_nat(C2),B) ) ) ).

% Diff_iff
tff(fact_2063_Diff__idemp,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B) ).

% Diff_idemp
tff(fact_2064_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2065_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2066_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2067_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2068_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_2069_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_zero
tff(fact_2070_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_2071_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_2072_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_2073_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_2074_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_2075_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_2076_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_2077_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% diff_self
tff(fact_2078_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_2079_Diff__empty,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_2080_Diff__empty,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),bot_bot_set_o) = A2 ).

% Diff_empty
tff(fact_2081_Diff__empty,axiom,
    ! [A2: set_set_nat] : minus_2163939370556025621et_nat(A2,bot_bot_set_set_nat) = A2 ).

% Diff_empty
tff(fact_2082_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_2083_Diff__empty,axiom,
    ! [A2: set_Extended_enat] : minus_925952699566721837d_enat(A2,bot_bo7653980558646680370d_enat) = A2 ).

% Diff_empty
tff(fact_2084_Diff__empty,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_2085_empty__Diff,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,bot_bot_set_int),A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_2086_empty__Diff,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,bot_bot_set_o),A2) = bot_bot_set_o ).

% empty_Diff
tff(fact_2087_empty__Diff,axiom,
    ! [A2: set_set_nat] : minus_2163939370556025621et_nat(bot_bot_set_set_nat,A2) = bot_bot_set_set_nat ).

% empty_Diff
tff(fact_2088_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_2089_empty__Diff,axiom,
    ! [A2: set_Extended_enat] : minus_925952699566721837d_enat(bot_bo7653980558646680370d_enat,A2) = bot_bo7653980558646680370d_enat ).

% empty_Diff
tff(fact_2090_empty__Diff,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_2091_Diff__cancel,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_2092_Diff__cancel,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),A2) = bot_bot_set_o ).

% Diff_cancel
tff(fact_2093_Diff__cancel,axiom,
    ! [A2: set_set_nat] : minus_2163939370556025621et_nat(A2,A2) = bot_bot_set_set_nat ).

% Diff_cancel
tff(fact_2094_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_2095_Diff__cancel,axiom,
    ! [A2: set_Extended_enat] : minus_925952699566721837d_enat(A2,A2) = bot_bo7653980558646680370d_enat ).

% Diff_cancel
tff(fact_2096_Diff__cancel,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_2097_finite__Diff2,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))
      <=> aa_set_int_o(finite_finite_int,A2) ) ) ).

% finite_Diff2
tff(fact_2098_finite__Diff2,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% finite_Diff2
tff(fact_2099_finite__Diff2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,B))
      <=> aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% finite_Diff2
tff(fact_2100_finite__Diff2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))
      <=> aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_Diff2
tff(fact_2101_finite__Diff,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) ) ).

% finite_Diff
tff(fact_2102_finite__Diff,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B)) ) ).

% finite_Diff
tff(fact_2103_finite__Diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,B)) ) ).

% finite_Diff
tff(fact_2104_finite__Diff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) ) ).

% finite_Diff
tff(fact_2105_Diff__insert0,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ( minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,B)) = minus_925952699566721837d_enat(A2,B) ) ) ).

% Diff_insert0
tff(fact_2106_Diff__insert0,axiom,
    ! [Xa: real,A2: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( minus_minus_set_real(A2,insert_real(Xa,B)) = minus_minus_set_real(A2,B) ) ) ).

% Diff_insert0
tff(fact_2107_Diff__insert0,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,B)) = minus_2163939370556025621et_nat(A2,B) ) ) ).

% Diff_insert0
tff(fact_2108_Diff__insert0,axiom,
    ! [Xa: int,A2: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B) ) ) ).

% Diff_insert0
tff(fact_2109_Diff__insert0,axiom,
    ! [Xa: $o,A2: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B) ) ) ).

% Diff_insert0
tff(fact_2110_Diff__insert0,axiom,
    ! [Xa: nat,A2: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B) ) ) ).

% Diff_insert0
tff(fact_2111_insert__Diff1,axiom,
    ! [Xa: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B)
     => ( minus_925952699566721837d_enat(insert_Extended_enat(Xa,A2),B) = minus_925952699566721837d_enat(A2,B) ) ) ).

% insert_Diff1
tff(fact_2112_insert__Diff1,axiom,
    ! [Xa: real,B: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),B)
     => ( minus_minus_set_real(insert_real(Xa,A2),B) = minus_minus_set_real(A2,B) ) ) ).

% insert_Diff1
tff(fact_2113_insert__Diff1,axiom,
    ! [Xa: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),B)
     => ( minus_2163939370556025621et_nat(insert_set_nat(Xa,A2),B) = minus_2163939370556025621et_nat(A2,B) ) ) ).

% insert_Diff1
tff(fact_2114_insert__Diff1,axiom,
    ! [Xa: int,B: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(Xa,A2)),B) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B) ) ) ).

% insert_Diff1
tff(fact_2115_insert__Diff1,axiom,
    ! [Xa: $o,B: set_o,A2: set_o] :
      ( aa_set_o_o(member_o((Xa)),B)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,insert_o((Xa),A2)),B) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B) ) ) ).

% insert_Diff1
tff(fact_2116_insert__Diff1,axiom,
    ! [Xa: nat,B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B) ) ) ).

% insert_Diff1
tff(fact_2117_Un__Diff__cancel,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),A2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) ).

% Un_Diff_cancel
tff(fact_2118_Un__Diff__cancel,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) ).

% Un_Diff_cancel
tff(fact_2119_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) ).

% Un_Diff_cancel
tff(fact_2120_Un__Diff__cancel2,axiom,
    ! [B: set_o,A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),A2)),A2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),A2) ).

% Un_Diff_cancel2
tff(fact_2121_Un__Diff__cancel2,axiom,
    ! [B: set_int,A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2)),A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),A2) ).

% Un_Diff_cancel2
tff(fact_2122_Un__Diff__cancel2,axiom,
    ! [B: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A2) ).

% Un_Diff_cancel2
tff(fact_2123_diff__ge__0__iff__ge,axiom,
    ! [A: 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,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2124_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2125_diff__ge__0__iff__ge,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2126_diff__gt__0__iff__gt,axiom,
    ! [A: 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,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2127_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2128_diff__gt__0__iff__gt,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2129_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_2130_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_2131_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2132_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_2133_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A2),B) ) ).

% Diff_eq_empty_iff
tff(fact_2134_Diff__eq__empty__iff,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( minus_2163939370556025621et_nat(A2,B) = bot_bot_set_set_nat )
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B) ) ).

% Diff_eq_empty_iff
tff(fact_2135_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( minus_minus_set_real(A2,B) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),B) ) ).

% Diff_eq_empty_iff
tff(fact_2136_Diff__eq__empty__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(A2,B) = bot_bo7653980558646680370d_enat )
    <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B) ) ).

% Diff_eq_empty_iff
tff(fact_2137_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ).

% Diff_eq_empty_iff
tff(fact_2138_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),B) ) ).

% Diff_eq_empty_iff
tff(fact_2139_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_2140_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),bot_bot_set_o))) = insert_o((A),A2) ).

% insert_Diff_single
tff(fact_2141_insert__Diff__single,axiom,
    ! [A: set_nat,A2: set_set_nat] : insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = insert_set_nat(A,A2) ).

% insert_Diff_single
tff(fact_2142_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_2143_insert__Diff__single,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] : insert_Extended_enat(A,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = insert_Extended_enat(A,A2) ).

% insert_Diff_single
tff(fact_2144_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_2145_finite__Diff__insert,axiom,
    ! [A2: set_o,A: $o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),B)))
    <=> aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)) ) ).

% finite_Diff_insert
tff(fact_2146_finite__Diff__insert,axiom,
    ! [A2: set_set_nat,A: set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,minus_2163939370556025621et_nat(A2,insert_set_nat(A,B)))
    <=> aa_set_set_nat_o(finite1152437895449049373et_nat,minus_2163939370556025621et_nat(A2,B)) ) ).

% finite_Diff_insert
tff(fact_2147_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B)))
    <=> aa_set_real_o(finite_finite_real,minus_minus_set_real(A2,B)) ) ).

% finite_Diff_insert
tff(fact_2148_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,B)))
    <=> aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) ) ).

% finite_Diff_insert
tff(fact_2149_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B)))
    <=> aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B)) ) ).

% finite_Diff_insert
tff(fact_2150_finite__Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B)))
    <=> aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A2,B)) ) ).

% finite_Diff_insert
tff(fact_2151_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B)))
    <=> aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) ) ).

% finite_Diff_insert
tff(fact_2152_Inf__fin_Osingleton,axiom,
    ! [Xa: nat] : lattic5238388535129920115in_nat(insert_nat(Xa,bot_bot_set_nat)) = Xa ).

% Inf_fin.singleton
tff(fact_2153_Inf__fin_Osingleton,axiom,
    ! [Xa: int] : lattic5235898064620869839in_int(insert_int(Xa,bot_bot_set_int)) = Xa ).

% Inf_fin.singleton
tff(fact_2154_Inf__fin_Osingleton,axiom,
    ! [Xa: $o] :
      ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((Xa),bot_bot_set_o))
    <=> (Xa) ) ).

% Inf_fin.singleton
tff(fact_2155_Inf__fin_Osingleton,axiom,
    ! [Xa: set_nat] : lattic3014633134055518761et_nat(insert_set_nat(Xa,bot_bot_set_set_nat)) = Xa ).

% Inf_fin.singleton
tff(fact_2156_Inf__fin_Osingleton,axiom,
    ! [Xa: real] : lattic2677971596711400399n_real(insert_real(Xa,bot_bot_set_real)) = Xa ).

% Inf_fin.singleton
tff(fact_2157_Inf__fin_Osingleton,axiom,
    ! [Xa: extended_enat] : lattic974744108425517955d_enat(insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = Xa ).

% Inf_fin.singleton
tff(fact_2158_sup__Inf__absorb,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,lattic2677971596711400399n_real(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2159_sup__Inf__absorb,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((A)),A2)
       => ( aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic4107685809792843317_fin_o,A2)),(A))
        <=> (A) ) ) ) ).

% sup_Inf_absorb
tff(fact_2160_sup__Inf__absorb,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic5235898064620869839in_int(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2161_sup__Inf__absorb,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => ( sup_su3973961784419623482d_enat(lattic974744108425517955d_enat(A2),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2162_sup__Inf__absorb,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3014633134055518761et_nat(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2163_sup__Inf__absorb,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic5238388535129920115in_nat(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2164_sup__Inf__absorb,axiom,
    ! [A2: set_set_o,A: set_o] :
      ( finite_finite_set_o(A2)
     => ( member_set_o(A,A2)
       => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,lattic3298725636695511317_set_o(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2165_sup__Inf__absorb,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,lattic8060154151401097861et_int(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2166_sup__Inf__absorb,axiom,
    ! [A2: set_filter_nat,A: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( member_filter_nat(A,A2)
       => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,lattic9128708996678779395er_nat(A2)),A) = A ) ) ) ).

% sup_Inf_absorb
tff(fact_2167_DiffE,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),minus_925952699566721837d_enat(A2,B))
     => ~ ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
         => aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% DiffE
tff(fact_2168_DiffE,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A2,B))
     => ~ ( aa_set_real_o(member_real(C2),A2)
         => aa_set_real_o(member_real(C2),B) ) ) ).

% DiffE
tff(fact_2169_DiffE,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A2,B))
     => ~ ( aa_set_set_nat_o(member_set_nat(C2),A2)
         => aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% DiffE
tff(fact_2170_DiffE,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))
     => ~ ( aa_set_int_o(member_int(C2),A2)
         => aa_set_int_o(member_int(C2),B) ) ) ).

% DiffE
tff(fact_2171_DiffE,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B))
     => ~ ( aa_set_o_o(member_o((C2)),A2)
         => aa_set_o_o(member_o((C2)),B) ) ) ).

% DiffE
tff(fact_2172_DiffE,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))
     => ~ ( aa_set_nat_o(member_nat(C2),A2)
         => aa_set_nat_o(member_nat(C2),B) ) ) ).

% DiffE
tff(fact_2173_DiffD1,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),minus_925952699566721837d_enat(A2,B))
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2) ) ).

% DiffD1
tff(fact_2174_DiffD1,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A2,B))
     => aa_set_real_o(member_real(C2),A2) ) ).

% DiffD1
tff(fact_2175_DiffD1,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A2,B))
     => aa_set_set_nat_o(member_set_nat(C2),A2) ) ).

% DiffD1
tff(fact_2176_DiffD1,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))
     => aa_set_int_o(member_int(C2),A2) ) ).

% DiffD1
tff(fact_2177_DiffD1,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B))
     => aa_set_o_o(member_o((C2)),A2) ) ).

% DiffD1
tff(fact_2178_DiffD1,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))
     => aa_set_nat_o(member_nat(C2),A2) ) ).

% DiffD1
tff(fact_2179_DiffD2,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),minus_925952699566721837d_enat(A2,B))
     => ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ).

% DiffD2
tff(fact_2180_DiffD2,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A2,B))
     => ~ aa_set_real_o(member_real(C2),B) ) ).

% DiffD2
tff(fact_2181_DiffD2,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A2,B))
     => ~ aa_set_set_nat_o(member_set_nat(C2),B) ) ).

% DiffD2
tff(fact_2182_DiffD2,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))
     => ~ aa_set_int_o(member_int(C2),B) ) ).

% DiffD2
tff(fact_2183_DiffD2,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B))
     => ~ aa_set_o_o(member_o((C2)),B) ) ).

% DiffD2
tff(fact_2184_DiffD2,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))
     => ~ aa_set_nat_o(member_nat(C2),B) ) ).

% DiffD2
tff(fact_2185_diff__eq__diff__eq,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( ( A = B2 )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2186_diff__eq__diff__eq,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),D) )
     => ( ( A = B2 )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2187_diff__eq__diff__eq,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( ( A = B2 )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2188_diff__right__commute,axiom,
    ! [A: real,C2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C2) ).

% diff_right_commute
tff(fact_2189_diff__right__commute,axiom,
    ! [A: rat,C2: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),B2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),C2) ).

% diff_right_commute
tff(fact_2190_diff__right__commute,axiom,
    ! [A: 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,A),C2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)),C2) ).

% diff_right_commute
tff(fact_2191_diff__right__commute,axiom,
    ! [A: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C2) ).

% diff_right_commute
tff(fact_2192_size__neq__size__imp__neq,axiom,
    ! [Xa: vEBT_VEBT,Ya: vEBT_VEBT] :
      ( ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Xa) != aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Ya) )
     => ( Xa != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_2193_size__neq__size__imp__neq,axiom,
    ! [Xa: num,Ya: num] :
      ( ( aa_num_nat(size_size_num,Xa) != aa_num_nat(size_size_num,Ya) )
     => ( Xa != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_2194_size__neq__size__imp__neq,axiom,
    ! [Xa: list_VEBT_VEBT,Ya: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xa) != size_s6755466524823107622T_VEBT(Ya) )
     => ( Xa != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_2195_size__neq__size__imp__neq,axiom,
    ! [Xa: list_o,Ya: list_o] :
      ( ( size_size_list_o(Xa) != size_size_list_o(Ya) )
     => ( Xa != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_2196_size__neq__size__imp__neq,axiom,
    ! [Xa: list_nat,Ya: list_nat] :
      ( ( size_size_list_nat(Xa) != size_size_list_nat(Ya) )
     => ( Xa != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_2197_diff__eq__diff__less__eq,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2198_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2199_diff__eq__diff__less__eq,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2200_diff__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) ) ).

% diff_right_mono
tff(fact_2201_diff__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),C2)) ) ).

% diff_right_mono
tff(fact_2202_diff__right__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) ) ).

% diff_right_mono
tff(fact_2203_diff__left__mono,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2)) ) ).

% diff_left_mono
tff(fact_2204_diff__left__mono,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B2)) ) ).

% diff_left_mono
tff(fact_2205_diff__left__mono,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2)) ) ).

% diff_left_mono
tff(fact_2206_diff__mono,axiom,
    ! [A: real,B2: real,D: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D)) ) ) ).

% diff_mono
tff(fact_2207_diff__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),D)) ) ) ).

% diff_mono
tff(fact_2208_diff__mono,axiom,
    ! [A: int,B2: int,D: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)) ) ) ).

% diff_mono
tff(fact_2209_eq__iff__diff__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_2210_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2211_eq__iff__diff__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_2212_diff__strict__mono,axiom,
    ! [A: real,B2: real,D: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D)) ) ) ).

% diff_strict_mono
tff(fact_2213_diff__strict__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),D)) ) ) ).

% diff_strict_mono
tff(fact_2214_diff__strict__mono,axiom,
    ! [A: int,B2: int,D: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)) ) ) ).

% diff_strict_mono
tff(fact_2215_diff__eq__diff__less,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_2216_diff__eq__diff__less,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_2217_diff__eq__diff__less,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_2218_diff__strict__left__mono,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2)) ) ).

% diff_strict_left_mono
tff(fact_2219_diff__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B2)) ) ).

% diff_strict_left_mono
tff(fact_2220_diff__strict__left__mono,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2)) ) ).

% diff_strict_left_mono
tff(fact_2221_diff__strict__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) ) ).

% diff_strict_right_mono
tff(fact_2222_diff__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),C2)) ) ).

% diff_strict_right_mono
tff(fact_2223_diff__strict__right__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) ) ).

% diff_strict_right_mono
tff(fact_2224_Diff__infinite__finite,axiom,
    ! [T2: set_int,S: set_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( ~ aa_set_int_o(finite_finite_int,S)
       => ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T2)) ) ) ).

% Diff_infinite_finite
tff(fact_2225_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_2226_Diff__infinite__finite,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_2227_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T2)) ) ) ).

% Diff_infinite_finite
tff(fact_2228_Diff__mono,axiom,
    ! [A2: set_nat,C: set_nat,D2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C),D2)) ) ) ).

% Diff_mono
tff(fact_2229_Diff__mono,axiom,
    ! [A2: set_int,C: set_int,D2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),C)
     => ( aa_set_int_o(ord_less_eq_set_int(D2),B)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C),D2)) ) ) ).

% Diff_mono
tff(fact_2230_Diff__subset,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),A2) ).

% Diff_subset
tff(fact_2231_Diff__subset,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),A2) ).

% Diff_subset
tff(fact_2232_double__diff,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_2233_double__diff,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_2234_insert__Diff__if,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      minus_925952699566721837d_enat(insert_Extended_enat(Xa,A2),B) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B),minus_925952699566721837d_enat(A2,B),insert_Extended_enat(Xa,minus_925952699566721837d_enat(A2,B))) ).

% insert_Diff_if
tff(fact_2235_insert__Diff__if,axiom,
    ! [Xa: real,A2: set_real,B: set_real] :
      minus_minus_set_real(insert_real(Xa,A2),B) = $ite(aa_set_real_o(member_real(Xa),B),minus_minus_set_real(A2,B),insert_real(Xa,minus_minus_set_real(A2,B))) ).

% insert_Diff_if
tff(fact_2236_insert__Diff__if,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B: set_set_nat] :
      minus_2163939370556025621et_nat(insert_set_nat(Xa,A2),B) = $ite(aa_set_set_nat_o(member_set_nat(Xa),B),minus_2163939370556025621et_nat(A2,B),insert_set_nat(Xa,minus_2163939370556025621et_nat(A2,B))) ).

% insert_Diff_if
tff(fact_2237_insert__Diff__if,axiom,
    ! [Xa: int,A2: set_int,B: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(Xa,A2)),B) = $ite(aa_set_int_o(member_int(Xa),B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B),insert_int(Xa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))) ).

% insert_Diff_if
tff(fact_2238_insert__Diff__if,axiom,
    ! [Xa: $o,A2: set_o,B: set_o] :
      aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,insert_o((Xa),A2)),B) = $ite(aa_set_o_o(member_o((Xa)),B),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B),insert_o((Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B))) ).

% insert_Diff_if
tff(fact_2239_insert__Diff__if,axiom,
    ! [Xa: nat,A2: set_nat,B: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),B) = $ite(aa_set_nat_o(member_nat(Xa),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B),insert_nat(Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))) ).

% insert_Diff_if
tff(fact_2240_Un__Diff,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),C) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),C)) ).

% Un_Diff
tff(fact_2241_Un__Diff,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),C) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),C)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),C)) ).

% Un_Diff
tff(fact_2242_Un__Diff,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),C)) ).

% Un_Diff
tff(fact_2243_psubset__imp__ex__mem,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),B)
     => ? [B4: extended_enat] : aa_set3337841544853333262enat_o(member_Extended_enat(B4),minus_925952699566721837d_enat(B,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2244_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B)
     => ? [B4: real] : aa_set_real_o(member_real(B4),minus_minus_set_real(B,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2245_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B)
     => ? [B4: set_nat] : aa_set_set_nat_o(member_set_nat(B4),minus_2163939370556025621et_nat(B,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2246_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B)
     => ? [B4: int] : aa_set_int_o(member_int(B4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2247_psubset__imp__ex__mem,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_set_o(A2),B)
     => ? [B4: $o] : aa_set_o_o(member_o((B4)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2248_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
     => ? [B4: nat] : aa_set_nat_o(member_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2249_le__iff__diff__le__0,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_2250_le__iff__diff__le__0,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_2251_le__iff__diff__le__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_2252_less__iff__diff__less__0,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_2253_less__iff__diff__less__0,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_2254_less__iff__diff__less__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_2255_diff__shunt__var,axiom,
    ! [Xa: set_o,Ya: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,Xa),Ya) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),Ya) ) ).

% diff_shunt_var
tff(fact_2256_diff__shunt__var,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] :
      ( ( minus_2163939370556025621et_nat(Xa,Ya) = bot_bot_set_set_nat )
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(Xa),Ya) ) ).

% diff_shunt_var
tff(fact_2257_diff__shunt__var,axiom,
    ! [Xa: set_real,Ya: set_real] :
      ( ( minus_minus_set_real(Xa,Ya) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xa),Ya) ) ).

% diff_shunt_var
tff(fact_2258_diff__shunt__var,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(Xa,Ya) = bot_bo7653980558646680370d_enat )
    <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(Xa),Ya) ) ).

% diff_shunt_var
tff(fact_2259_diff__shunt__var,axiom,
    ! [Xa: set_nat,Ya: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Xa),Ya) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),Ya) ) ).

% diff_shunt_var
tff(fact_2260_diff__shunt__var,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Xa),Ya) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),Ya) ) ).

% diff_shunt_var
tff(fact_2261_Diff__insert,axiom,
    ! [A2: set_int,A: int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_2262_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)),insert_o((A),bot_bot_set_o)) ).

% Diff_insert
tff(fact_2263_Diff__insert,axiom,
    ! [A2: set_set_nat,A: set_nat,B: set_set_nat] : minus_2163939370556025621et_nat(A2,insert_set_nat(A,B)) = minus_2163939370556025621et_nat(minus_2163939370556025621et_nat(A2,B),insert_set_nat(A,bot_bot_set_set_nat)) ).

% Diff_insert
tff(fact_2264_Diff__insert,axiom,
    ! [A2: set_real,A: real,B: set_real] : minus_minus_set_real(A2,insert_real(A,B)) = minus_minus_set_real(minus_minus_set_real(A2,B),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_2265_Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] : minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B)) = minus_925952699566721837d_enat(minus_925952699566721837d_enat(A2,B),insert_Extended_enat(A,bot_bo7653980558646680370d_enat)) ).

% Diff_insert
tff(fact_2266_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_2267_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_2268_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),bot_bot_set_o))) = A2 ) ) ).

% insert_Diff
tff(fact_2269_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_2270_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_2271_insert__Diff,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ( insert_Extended_enat(A,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = A2 ) ) ).

% insert_Diff
tff(fact_2272_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_2273_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,bot_bot_set_int))),B) ).

% Diff_insert2
tff(fact_2274_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),bot_bot_set_o))),B) ).

% Diff_insert2
tff(fact_2275_Diff__insert2,axiom,
    ! [A2: set_set_nat,A: set_nat,B: set_set_nat] : minus_2163939370556025621et_nat(A2,insert_set_nat(A,B)) = minus_2163939370556025621et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat)),B) ).

% Diff_insert2
tff(fact_2276_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B: set_real] : minus_minus_set_real(A2,insert_real(A,B)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B) ).

% Diff_insert2
tff(fact_2277_Diff__insert2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] : minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B)) = minus_925952699566721837d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)),B) ).

% Diff_insert2
tff(fact_2278_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,bot_bot_set_nat))),B) ).

% Diff_insert2
tff(fact_2279_Diff__insert__absorb,axiom,
    ! [Xa: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(Xa,A2)),insert_int(Xa,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2280_Diff__insert__absorb,axiom,
    ! [Xa: $o,A2: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,insert_o((Xa),A2)),insert_o((Xa),bot_bot_set_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2281_Diff__insert__absorb,axiom,
    ! [Xa: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( minus_2163939370556025621et_nat(insert_set_nat(Xa,A2),insert_set_nat(Xa,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2282_Diff__insert__absorb,axiom,
    ! [Xa: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( minus_minus_set_real(insert_real(Xa,A2),insert_real(Xa,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2283_Diff__insert__absorb,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ( minus_925952699566721837d_enat(insert_Extended_enat(Xa,A2),insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2284_Diff__insert__absorb,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(Xa,A2)),insert_nat(Xa,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2285_subset__Diff__insert,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,Xa: extended_enat,C: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),minus_925952699566721837d_enat(B,insert_Extended_enat(Xa,C)))
    <=> ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),minus_925952699566721837d_enat(B,C))
        & ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_2286_subset__Diff__insert,axiom,
    ! [A2: set_real,B: set_real,Xa: real,C: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B,insert_real(Xa,C)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B,C))
        & ~ aa_set_real_o(member_real(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_2287_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B: set_set_nat,Xa: set_nat,C: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B,insert_set_nat(Xa,C)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B,C))
        & ~ aa_set_set_nat_o(member_set_nat(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_2288_subset__Diff__insert,axiom,
    ! [A2: set_o,B: set_o,Xa: $o,C: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),insert_o((Xa),C)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),C))
        & ~ aa_set_o_o(member_o((Xa)),A2) ) ) ).

% subset_Diff_insert
tff(fact_2289_subset__Diff__insert,axiom,
    ! [A2: set_nat,B: set_nat,Xa: nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),insert_nat(Xa,C)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),C))
        & ~ aa_set_nat_o(member_nat(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_2290_subset__Diff__insert,axiom,
    ! [A2: set_int,B: set_int,Xa: int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),insert_int(Xa,C)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),C))
        & ~ aa_set_int_o(member_int(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_2291_Diff__subset__conv,axiom,
    ! [A2: set_o,B: set_o,C: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)),C)
    <=> aa_set_o_o(ord_less_eq_set_o(A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) ) ).

% Diff_subset_conv
tff(fact_2292_Diff__subset__conv,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),C)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ) ).

% Diff_subset_conv
tff(fact_2293_Diff__subset__conv,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),C)
    <=> aa_set_int_o(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) ) ).

% Diff_subset_conv
tff(fact_2294_Diff__partition,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),A2)) = B ) ) ).

% Diff_partition
tff(fact_2295_Diff__partition,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2)) = B ) ) ).

% Diff_partition
tff(fact_2296_Diff__partition,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2)) = B ) ) ).

% Diff_partition
tff(fact_2297_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic2677971596711400399n_real(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2298_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2299_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((A)),A2)
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,A2)),(A)) ) ) ).

% Inf_fin.coboundedI
tff(fact_2300_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic974744108425517955d_enat(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2301_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => aa_set_int_o(ord_less_eq_set_int(lattic8060154151401097861et_int(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2302_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic4603258475043424379in_rat(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2303_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic5238388535129920115in_nat(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2304_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(A2)),A) ) ) ).

% Inf_fin.coboundedI
tff(fact_2305_infinite__remove,axiom,
    ! [S: set_complex,A: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S,insert_complex(A,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_2306_infinite__remove,axiom,
    ! [S: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_2307_infinite__remove,axiom,
    ! [S: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ~ aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,S),insert_o((A),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_2308_infinite__remove,axiom,
    ! [S: set_set_nat,A: set_nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,minus_2163939370556025621et_nat(S,insert_set_nat(A,bot_bot_set_set_nat))) ) ).

% infinite_remove
tff(fact_2309_infinite__remove,axiom,
    ! [S: set_real,A: real] :
      ( ~ aa_set_real_o(finite_finite_real,S)
     => ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(S,insert_real(A,bot_bot_set_real))) ) ).

% infinite_remove
tff(fact_2310_infinite__remove,axiom,
    ! [S: set_Extended_enat,A: extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ).

% infinite_remove
tff(fact_2311_infinite__remove,axiom,
    ! [S: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_2312_infinite__coinduct,axiom,
    ! [X3: fun_set_complex_o,A2: set_complex] :
      ( aa_set_complex_o(X3,A2)
     => ( ! [A5: set_complex] :
            ( aa_set_complex_o(X3,A5)
           => ? [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),A5)
                & ( aa_set_complex_o(X3,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% infinite_coinduct
tff(fact_2313_infinite__coinduct,axiom,
    ! [X3: fun_set_int_o,A2: set_int] :
      ( aa_set_int_o(X3,A2)
     => ( ! [A5: set_int] :
            ( aa_set_int_o(X3,A5)
           => ? [X4: int] :
                ( aa_set_int_o(member_int(X4),A5)
                & ( aa_set_int_o(X3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A5),insert_int(X4,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A5),insert_int(X4,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% infinite_coinduct
tff(fact_2314_infinite__coinduct,axiom,
    ! [X3: fun_set_o_o,A2: set_o] :
      ( aa_set_o_o(X3,A2)
     => ( ! [A5: set_o] :
            ( aa_set_o_o(X3,A5)
           => ? [X4: $o] :
                ( aa_set_o_o(member_o((X4)),A5)
                & ( aa_set_o_o(X3,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A5),insert_o((X4),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A5),insert_o((X4),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% infinite_coinduct
tff(fact_2315_infinite__coinduct,axiom,
    ! [X3: fun_set_set_nat_o,A2: set_set_nat] :
      ( aa_set_set_nat_o(X3,A2)
     => ( ! [A5: set_set_nat] :
            ( aa_set_set_nat_o(X3,A5)
           => ? [X4: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(X4),A5)
                & ( aa_set_set_nat_o(X3,minus_2163939370556025621et_nat(A5,insert_set_nat(X4,bot_bot_set_set_nat)))
                  | ~ aa_set_set_nat_o(finite1152437895449049373et_nat,minus_2163939370556025621et_nat(A5,insert_set_nat(X4,bot_bot_set_set_nat))) ) ) )
       => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_2316_infinite__coinduct,axiom,
    ! [X3: fun_set_real_o,A2: set_real] :
      ( aa_set_real_o(X3,A2)
     => ( ! [A5: set_real] :
            ( aa_set_real_o(X3,A5)
           => ? [X4: real] :
                ( aa_set_real_o(member_real(X4),A5)
                & ( aa_set_real_o(X3,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real)))
                  | ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real))) ) ) )
       => ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% infinite_coinduct
tff(fact_2317_infinite__coinduct,axiom,
    ! [X3: fun_se8123728892507317397enat_o,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(X3,A2)
     => ( ! [A5: set_Extended_enat] :
            ( aa_set3337841544853333262enat_o(X3,A5)
           => ? [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A5)
                & ( aa_set3337841544853333262enat_o(X3,minus_925952699566721837d_enat(A5,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat)))
                  | ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A5,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat))) ) ) )
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% infinite_coinduct
tff(fact_2318_infinite__coinduct,axiom,
    ! [X3: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X3,A2)
     => ( ! [A5: set_nat] :
            ( aa_set_nat_o(X3,A5)
           => ? [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),A5)
                & ( aa_set_nat_o(X3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),insert_nat(X4,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),insert_nat(X4,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_2319_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(P,A2)
       => ( ! [A3: complex,A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( aa_set_complex_o(member_complex(A3),A5)
               => ( aa_set_complex_o(P,A5)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_2320_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,A2)
       => ( ! [A3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( aa_set_int_o(member_int(A3),A5)
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A5),insert_int(A3,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_2321_finite__empty__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,A2)
       => ( ! [A3: $o,A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( aa_set_o_o(member_o((A3)),A5)
               => ( aa_set_o_o(P,A5)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A5),insert_o((A3),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_2322_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(P,A2)
       => ( ! [A3: set_nat,A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( aa_set_set_nat_o(member_set_nat(A3),A5)
               => ( aa_set_set_nat_o(P,A5)
                 => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(A3,bot_bot_set_set_nat))) ) ) )
         => aa_set_set_nat_o(P,bot_bot_set_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_2323_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,A2)
       => ( ! [A3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( aa_set_real_o(member_real(A3),A5)
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(A3,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_2324_finite__empty__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,A2)
       => ( ! [A3: extended_enat,A5: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
             => ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A5)
               => ( aa_set3337841544853333262enat_o(P,A5)
                 => aa_set3337841544853333262enat_o(P,minus_925952699566721837d_enat(A5,insert_Extended_enat(A3,bot_bo7653980558646680370d_enat))) ) ) )
         => aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat) ) ) ) ).

% finite_empty_induct
tff(fact_2325_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,A2)
       => ( ! [A3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( aa_set_nat_o(member_nat(A3),A5)
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),insert_nat(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_2326_subset__insert__iff,axiom,
    ! [A2: set_o,Xa: $o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),B))
    <=> $ite(aa_set_o_o(member_o((Xa)),A2),aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))),B),aa_set_o_o(ord_less_eq_set_o(A2),B)) ) ).

% subset_insert_iff
tff(fact_2327_subset__insert__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(Xa,B))
    <=> $ite(aa_set_set_nat_o(member_set_nat(Xa),A2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))),B),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)) ) ).

% subset_insert_iff
tff(fact_2328_subset__insert__iff,axiom,
    ! [A2: set_real,Xa: real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B))
    <=> $ite(aa_set_real_o(member_real(Xa),A2),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B),aa_set_real_o(ord_less_eq_set_real(A2),B)) ) ).

% subset_insert_iff
tff(fact_2329_subset__insert__iff,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(Xa,B))
    <=> $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2),aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))),B),aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)) ) ).

% subset_insert_iff
tff(fact_2330_subset__insert__iff,axiom,
    ! [A2: set_nat,Xa: nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(Xa,B))
    <=> $ite(aa_set_nat_o(member_nat(Xa),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),B),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)) ) ).

% subset_insert_iff
tff(fact_2331_subset__insert__iff,axiom,
    ! [A2: set_int,Xa: int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B))
    <=> $ite(aa_set_int_o(member_int(Xa),A2),aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))),B),aa_set_int_o(ord_less_eq_set_int(A2),B)) ) ).

% subset_insert_iff
tff(fact_2332_Diff__single__insert,axiom,
    ! [A2: set_o,Xa: $o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))),B)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),B)) ) ).

% Diff_single_insert
tff(fact_2333_Diff__single__insert,axiom,
    ! [A2: set_set_nat,Xa: set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))),B)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(Xa,B)) ) ).

% Diff_single_insert
tff(fact_2334_Diff__single__insert,axiom,
    ! [A2: set_real,Xa: real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B)) ) ).

% Diff_single_insert
tff(fact_2335_Diff__single__insert,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))),B)
     => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),insert_Extended_enat(Xa,B)) ) ).

% Diff_single_insert
tff(fact_2336_Diff__single__insert,axiom,
    ! [A2: set_nat,Xa: nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(Xa,B)) ) ).

% Diff_single_insert
tff(fact_2337_Diff__single__insert,axiom,
    ! [A2: set_int,Xa: int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))),B)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B)) ) ).

% Diff_single_insert
tff(fact_2338_pinf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q2,X2) ) )
       => ? [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_2339_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q2,X2) ) )
       => ? [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_2340_pinf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q2,X2) ) )
       => ? [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_2341_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q2,X2) ) )
       => ? [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_2342_pinf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q2,X2) ) )
       => ? [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_2343_pinf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q2,X2) ) )
       => ? [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_2344_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q2,X2) ) )
       => ? [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_2345_pinf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q2,X2) ) )
       => ? [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_2346_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q2,X2) ) )
       => ? [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_2347_pinf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q2,X2) ) )
       => ? [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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_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_2356_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_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_minf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q2,X2) ) )
       => ? [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_2369_minf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q2,X2) ) )
       => ? [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_2370_minf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z3)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z3)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q2,X2) ) )
       => ? [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_2371_minf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q2,X2) ) )
       => ? [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_2372_minf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q2,X2) ) )
       => ? [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_2373_minf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q2,X2) ) )
       => ? [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_2374_minf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q2,X2) ) )
       => ? [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_2375_minf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z3)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z3)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q2,X2) ) )
       => ? [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_2376_minf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q2,X2) ) )
       => ? [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_2377_minf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q2,X2) ) )
       => ? [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_2378_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_2379_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_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_Inf__fin_OboundedE,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
         => ! [A8: $o] :
              ( aa_set_o_o(member_o((A8)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(A8)) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2399_Inf__fin_OboundedE,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),lattic3014633134055518761et_nat(A2))
         => ! [A8: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(A8),A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2400_Inf__fin_OboundedE,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic2677971596711400399n_real(A2))
         => ! [A8: real] :
              ( aa_set_real_o(member_real(A8),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2401_Inf__fin_OboundedE,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),lattic974744108425517955d_enat(A2))
         => ! [A8: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A8),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2402_Inf__fin_OboundedE,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(Xa),lattic8060154151401097861et_int(A2))
         => ! [A8: set_int] :
              ( member_set_int(A8,A2)
             => aa_set_int_o(ord_less_eq_set_int(Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2403_Inf__fin_OboundedE,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic4603258475043424379in_rat(A2))
         => ! [A8: rat] :
              ( aa_set_rat_o(member_rat(A8),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2404_Inf__fin_OboundedE,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),lattic5238388535129920115in_nat(A2))
         => ! [A8: nat] :
              ( aa_set_nat_o(member_nat(A8),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2405_Inf__fin_OboundedE,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic5235898064620869839in_int(A2))
         => ! [A8: int] :
              ( aa_set_int_o(member_int(A8),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A8) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2406_Inf__fin_OboundedI,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(A3)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2407_Inf__fin_OboundedI,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(A3),A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),A3) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),lattic3014633134055518761et_nat(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2408_Inf__fin_OboundedI,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A3) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic2677971596711400399n_real(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2409_Inf__fin_OboundedI,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [A3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),A3) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),lattic974744108425517955d_enat(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2410_Inf__fin_OboundedI,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( ! [A3: set_int] :
              ( member_set_int(A3,A2)
             => aa_set_int_o(ord_less_eq_set_int(Xa),A3) )
         => aa_set_int_o(ord_less_eq_set_int(Xa),lattic8060154151401097861et_int(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2411_Inf__fin_OboundedI,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A3) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic4603258475043424379in_rat(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2412_Inf__fin_OboundedI,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A3) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),lattic5238388535129920115in_nat(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2413_Inf__fin_OboundedI,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A3) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic5235898064620869839in_int(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2414_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
        <=> ! [X: $o] :
              ( aa_set_o_o(member_o((X)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X)) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2415_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),lattic3014633134055518761et_nat(A2))
        <=> ! [X: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(X),A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2416_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic2677971596711400399n_real(A2))
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2417_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),lattic974744108425517955d_enat(A2))
        <=> ! [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2418_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_set_int,Xa: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(Xa),lattic8060154151401097861et_int(A2))
        <=> ! [X: set_int] :
              ( member_set_int(X,A2)
             => aa_set_int_o(ord_less_eq_set_int(Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2419_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic4603258475043424379in_rat(A2))
        <=> ! [X: rat] :
              ( aa_set_rat_o(member_rat(X),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2420_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),lattic5238388535129920115in_nat(A2))
        <=> ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2421_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic5235898064620869839in_int(A2))
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2422_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) )
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B)
                 => ( ! [X4: complex] :
                        ( aa_set_complex_o(member_complex(X4),A5)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A5) ) ) ) )
         => aa_set_complex_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2423_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) )
       => ( ! [A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ( A5 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A5),B)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A5)
                       => aa_set_o_o(P,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A5),insert_o((X4),bot_bot_set_o))) )
                   => aa_set_o_o(P,A5) ) ) ) )
         => aa_set_o_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2424_remove__induct,axiom,
    ! [P: fun_set_set_nat_o,B: set_set_nat] :
      ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B)
         => aa_set_set_nat_o(P,B) )
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B)
                 => ( ! [X4: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X4),A5)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X4,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A5) ) ) ) )
         => aa_set_set_nat_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2425_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) )
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B)
                 => ( ! [X4: real] :
                        ( aa_set_real_o(member_real(X4),A5)
                       => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real))) )
                   => aa_set_real_o(P,A5) ) ) ) )
         => aa_set_real_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2426_remove__induct,axiom,
    ! [P: fun_se8123728892507317397enat_o,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
     => ( ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
         => aa_set3337841544853333262enat_o(P,B) )
       => ( ! [A5: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
             => ( ( A5 != bot_bo7653980558646680370d_enat )
               => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A5),B)
                 => ( ! [X4: extended_enat] :
                        ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A5)
                       => aa_set3337841544853333262enat_o(P,minus_925952699566721837d_enat(A5,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat))) )
                   => aa_set3337841544853333262enat_o(P,A5) ) ) ) )
         => aa_set3337841544853333262enat_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2427_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) )
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),B)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat(X4),A5)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),insert_nat(X4,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A5) ) ) ) )
         => aa_set_nat_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2428_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) )
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B)
                 => ( ! [X4: int] :
                        ( aa_set_int_o(member_int(X4),A5)
                       => aa_set_int_o(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A5),insert_int(X4,bot_bot_set_int))) )
                   => aa_set_int_o(P,A5) ) ) ) )
         => aa_set_int_o(P,B) ) ) ) ).

% remove_induct
tff(fact_2429_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)
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B)
                 => ( ! [X4: complex] :
                        ( aa_set_complex_o(member_complex(X4),A5)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A5) ) ) ) )
         => aa_set_complex_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2430_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)
       => ( ! [A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ( A5 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A5),B)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A5)
                       => aa_set_o_o(P,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A5),insert_o((X4),bot_bot_set_o))) )
                   => aa_set_o_o(P,A5) ) ) ) )
         => aa_set_o_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2431_finite__remove__induct,axiom,
    ! [B: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B)
                 => ( ! [X4: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X4),A5)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X4,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A5) ) ) ) )
         => aa_set_set_nat_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2432_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)
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B)
                 => ( ! [X4: real] :
                        ( aa_set_real_o(member_real(X4),A5)
                       => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real))) )
                   => aa_set_real_o(P,A5) ) ) ) )
         => aa_set_real_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2433_finite__remove__induct,axiom,
    ! [B: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [A5: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A5)
             => ( ( A5 != bot_bo7653980558646680370d_enat )
               => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A5),B)
                 => ( ! [X4: extended_enat] :
                        ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A5)
                       => aa_set3337841544853333262enat_o(P,minus_925952699566721837d_enat(A5,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat))) )
                   => aa_set3337841544853333262enat_o(P,A5) ) ) ) )
         => aa_set3337841544853333262enat_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2434_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)
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),B)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat(X4),A5)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),insert_nat(X4,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A5) ) ) ) )
         => aa_set_nat_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2435_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)
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B)
                 => ( ! [X4: int] :
                        ( aa_set_int_o(member_int(X4),A5)
                       => aa_set_int_o(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A5),insert_int(X4,bot_bot_set_int))) )
                   => aa_set_int_o(P,A5) ) ) ) )
         => aa_set_int_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_2436_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)
       => ( ! [T3: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T3),S)
             => ( aa_set_complex_o(P,T3)
               => ? [X4: complex] :
                    ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(S,T3))
                    & aa_set_complex_o(P,insert_complex(X4,T3)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2437_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)
       => ( ! [T3: set_int] :
              ( aa_set_int_o(ord_less_set_int(T3),S)
             => ( aa_set_int_o(P,T3)
               => ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T3))
                    & aa_set_int_o(P,insert_int(X4,T3)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2438_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)
       => ( ! [T3: set_o] :
              ( aa_set_o_o(ord_less_set_o(T3),S)
             => ( aa_set_o_o(P,T3)
               => ? [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,S),T3))
                    & aa_set_o_o(P,insert_o((X4),T3)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2439_finite__induct__select,axiom,
    ! [S: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [T3: set_set_nat] :
              ( aa_set_set_nat_o(ord_less_set_set_nat(T3),S)
             => ( aa_set_set_nat_o(P,T3)
               => ? [X4: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat(X4),minus_2163939370556025621et_nat(S,T3))
                    & aa_set_set_nat_o(P,insert_set_nat(X4,T3)) ) ) )
         => aa_set_set_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2440_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)
       => ( ! [T3: set_real] :
              ( aa_set_real_o(ord_less_set_real(T3),S)
             => ( aa_set_real_o(P,T3)
               => ? [X4: real] :
                    ( aa_set_real_o(member_real(X4),minus_minus_set_real(S,T3))
                    & aa_set_real_o(P,insert_real(X4,T3)) ) ) )
         => aa_set_real_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2441_finite__induct__select,axiom,
    ! [S: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [T3: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(T3),S)
             => ( aa_set3337841544853333262enat_o(P,T3)
               => ? [X4: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(S,T3))
                    & aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,T3)) ) ) )
         => aa_set3337841544853333262enat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2442_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)
       => ( ! [T3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T3),S)
             => ( aa_set_nat_o(P,T3)
               => ? [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T3))
                    & aa_set_nat_o(P,insert_nat(X4,T3)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_2443_psubset__insert__iff,axiom,
    ! [A2: set_o,Xa: $o,B: set_o] :
      ( aa_set_o_o(ord_less_set_o(A2),insert_o((Xa),B))
    <=> $ite(
          aa_set_o_o(member_o((Xa)),B),
          aa_set_o_o(ord_less_set_o(A2),B),
          $ite(aa_set_o_o(member_o((Xa)),A2),aa_set_o_o(ord_less_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))),B),aa_set_o_o(ord_less_eq_set_o(A2),B)) ) ) ).

% psubset_insert_iff
tff(fact_2444_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),insert_set_nat(Xa,B))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(Xa),B),
          aa_set_set_nat_o(ord_less_set_set_nat(A2),B),
          $ite(aa_set_set_nat_o(member_set_nat(Xa),A2),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))),B),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)) ) ) ).

% psubset_insert_iff
tff(fact_2445_psubset__insert__iff,axiom,
    ! [A2: set_real,Xa: real,B: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real(Xa,B))
    <=> $ite(
          aa_set_real_o(member_real(Xa),B),
          aa_set_real_o(ord_less_set_real(A2),B),
          $ite(aa_set_real_o(member_real(Xa),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B),aa_set_real_o(ord_less_eq_set_real(A2),B)) ) ) ).

% psubset_insert_iff
tff(fact_2446_psubset__insert__iff,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),insert_Extended_enat(Xa,B))
    <=> $ite(
          aa_set3337841544853333262enat_o(member_Extended_enat(Xa),B),
          aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),B),
          $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2),aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))),B),aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)) ) ) ).

% psubset_insert_iff
tff(fact_2447_psubset__insert__iff,axiom,
    ! [A2: set_nat,Xa: nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),insert_nat(Xa,B))
    <=> $ite(
          aa_set_nat_o(member_nat(Xa),B),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B),
          $ite(aa_set_nat_o(member_nat(Xa),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),B),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)) ) ) ).

% psubset_insert_iff
tff(fact_2448_psubset__insert__iff,axiom,
    ! [A2: set_int,Xa: int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int(Xa,B))
    <=> $ite(
          aa_set_int_o(member_int(Xa),B),
          aa_set_int_o(ord_less_set_int(A2),B),
          $ite(aa_set_int_o(member_int(Xa),A2),aa_set_int_o(ord_less_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))),B),aa_set_int_o(ord_less_eq_set_int(A2),B)) ) ) ).

% psubset_insert_iff
tff(fact_2449_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,B)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2450_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(B)),lattic3014633134055518761et_nat(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2451_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_real_o(finite_finite_real,B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic2677971596711400399n_real(B)),lattic2677971596711400399n_real(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2452_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,lattic974744108425517955d_enat(B)),lattic974744108425517955d_enat(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2453_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_set_int,B: set_set_int] :
      ( ord_le4403425263959731960et_int(A2,B)
     => ( ( A2 != bot_bot_set_set_int )
       => ( finite6197958912794628473et_int(B)
         => aa_set_int_o(ord_less_eq_set_int(lattic8060154151401097861et_int(B)),lattic8060154151401097861et_int(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2454_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_rat,B: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A2),B)
     => ( ( A2 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic4603258475043424379in_rat(B)),lattic4603258475043424379in_rat(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2455_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic5238388535129920115in_nat(B)),lattic5238388535129920115in_nat(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2456_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(B)),lattic5235898064620869839in_int(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2457_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_2458_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_2459_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_2460_remove__def,axiom,
    ! [Xa: int,A2: set_int] : remove_int(Xa,A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)) ).

% remove_def
tff(fact_2461_remove__def,axiom,
    ! [Xa: $o,A2: set_o] : remove_o((Xa),A2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)) ).

% remove_def
tff(fact_2462_remove__def,axiom,
    ! [Xa: set_nat,A2: set_set_nat] : remove_set_nat(Xa,A2) = minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)) ).

% remove_def
tff(fact_2463_remove__def,axiom,
    ! [Xa: real,A2: set_real] : remove_real(Xa,A2) = minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)) ).

% remove_def
tff(fact_2464_remove__def,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat] : remove_Extended_enat(Xa,A2) = minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) ).

% remove_def
tff(fact_2465_remove__def,axiom,
    ! [Xa: nat,A2: set_nat] : remove_nat(Xa,A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)) ).

% remove_def
tff(fact_2466_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_se7855581050983116737at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( lattic30941717366863870at_nat(insert9200635055090092081at_nat(Xa,A2)) = $ite(minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat)) = bot_bo3083307316010499117at_nat,Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),lattic30941717366863870at_nat(minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2467_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( lattic5235898064620869839in_int(insert_int(Xa,A2)) = $ite(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)) = bot_bot_set_int,Xa,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),lattic5235898064620869839in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2468_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((Xa),A2))
      <=> $ite(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)) = bot_bot_set_o,(Xa),aa_o_o(aa_o_fun_o_o(inf_inf_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2469_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( lattic3014633134055518761et_nat(insert_set_nat(Xa,A2)) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)) = bot_bot_set_set_nat,Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),lattic3014633134055518761et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2470_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( lattic2677971596711400399n_real(insert_real(Xa,A2)) = $ite(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)) = bot_bot_set_real,Xa,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,Xa),lattic2677971596711400399n_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2471_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( lattic974744108425517955d_enat(insert_Extended_enat(Xa,A2)) = $ite(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = bot_bo7653980558646680370d_enat,Xa,inf_in1870772243966228564d_enat(Xa,lattic974744108425517955d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2472_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( lattic5238388535129920115in_nat(insert_nat(Xa,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)) = bot_bot_set_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),lattic5238388535129920115in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2473_Inf__fin_Oremove,axiom,
    ! [A2: set_se7855581050983116737at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( member2643936169264416010at_nat(Xa,A2)
       => ( lattic30941717366863870at_nat(A2) = $ite(minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat)) = bot_bo3083307316010499117at_nat,Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),lattic30941717366863870at_nat(minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2474_Inf__fin_Oremove,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( lattic5235898064620869839in_int(A2) = $ite(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)) = bot_bot_set_int,Xa,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),lattic5235898064620869839in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2475_Inf__fin_Oremove,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,A2)
        <=> $ite(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)) = bot_bot_set_o,(Xa),aa_o_o(aa_o_fun_o_o(inf_inf_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2476_Inf__fin_Oremove,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( lattic3014633134055518761et_nat(A2) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)) = bot_bot_set_set_nat,Xa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),lattic3014633134055518761et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2477_Inf__fin_Oremove,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( lattic2677971596711400399n_real(A2) = $ite(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)) = bot_bot_set_real,Xa,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,Xa),lattic2677971596711400399n_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2478_Inf__fin_Oremove,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( lattic974744108425517955d_enat(A2) = $ite(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) = bot_bo7653980558646680370d_enat,Xa,inf_in1870772243966228564d_enat(Xa,lattic974744108425517955d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2479_Inf__fin_Oremove,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( lattic5238388535129920115in_nat(A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)) = bot_bot_set_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),lattic5238388535129920115in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2480_card__Diff1__less__iff,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A2))
    <=> ( finite4290736615968046902t_unit(A2)
        & member_Product_unit(Xa,A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2481_card__Diff1__less__iff,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)))),finite_card_list_nat(A2))
    <=> ( finite8100373058378681591st_nat(A2)
        & member_list_nat(Xa,A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2482_card__Diff1__less__iff,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))),finite_card_complex(A2))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,A2)
        & aa_set_complex_o(member_complex(Xa),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2483_card__Diff1__less__iff,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)))),finite_card_int(A2))
    <=> ( aa_set_int_o(finite_finite_int,A2)
        & aa_set_int_o(member_int(Xa),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2484_card__Diff1__less__iff,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)))),finite_card_o(A2))
    <=> ( aa_set_o_o(finite_finite_o,A2)
        & aa_set_o_o(member_o((Xa)),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2485_card__Diff1__less__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)))),finite_card_set_nat(A2))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
        & aa_set_set_nat_o(member_set_nat(Xa),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2486_card__Diff1__less__iff,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))),finite_card_real(A2))
    <=> ( aa_set_real_o(finite_finite_real,A2)
        & aa_set_real_o(member_real(Xa),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2487_card__Diff1__less__iff,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A2))
    <=> ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
        & aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2488_card__Diff1__less__iff,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)))),finite_card_nat(A2))
    <=> ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(Xa),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_2489_card__Diff2__less,axiom,
    ! [A2: set_Product_unit,Xa: product_unit,Ya: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(Xa,A2)
       => ( member_Product_unit(Ya,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)),insert_Product_unit(Ya,bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2490_card__Diff2__less,axiom,
    ! [A2: set_list_nat,Xa: list_nat,Ya: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( member_list_nat(Xa,A2)
       => ( member_list_nat(Ya,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)),insert_list_nat(Ya,bot_bot_set_list_nat)))),finite_card_list_nat(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2491_card__Diff2__less,axiom,
    ! [A2: set_complex,Xa: complex,Ya: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( aa_set_complex_o(member_complex(Ya),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)),insert_complex(Ya,bot_bot_set_complex)))),finite_card_complex(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2492_card__Diff2__less,axiom,
    ! [A2: set_int,Xa: int,Ya: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( aa_set_int_o(member_int(Ya),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))),insert_int(Ya,bot_bot_set_int)))),finite_card_int(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2493_card__Diff2__less,axiom,
    ! [A2: set_o,Xa: $o,Ya: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( aa_set_o_o(member_o((Ya)),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))),insert_o((Ya),bot_bot_set_o)))),finite_card_o(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2494_card__Diff2__less,axiom,
    ! [A2: set_set_nat,Xa: set_nat,Ya: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( aa_set_set_nat_o(member_set_nat(Ya),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)),insert_set_nat(Ya,bot_bot_set_set_nat)))),finite_card_set_nat(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2495_card__Diff2__less,axiom,
    ! [A2: set_real,Xa: real,Ya: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( aa_set_real_o(member_real(Ya),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_real(minus_minus_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)),insert_real(Ya,bot_bot_set_real)))),finite_card_real(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2496_card__Diff2__less,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat,Ya: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ya),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)),insert_Extended_enat(Ya,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2497_card__Diff2__less,axiom,
    ! [A2: set_nat,Xa: nat,Ya: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( aa_set_nat_o(member_nat(Ya),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))),insert_nat(Ya,bot_bot_set_nat)))),finite_card_nat(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_2498_card__Diff1__less,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(Xa,A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A2)) ) ) ).

% card_Diff1_less
tff(fact_2499_card__Diff1__less,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( member_list_nat(Xa,A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)))),finite_card_list_nat(A2)) ) ) ).

% card_Diff1_less
tff(fact_2500_card__Diff1__less,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))),finite_card_complex(A2)) ) ) ).

% card_Diff1_less
tff(fact_2501_card__Diff1__less,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)))),finite_card_int(A2)) ) ) ).

% card_Diff1_less
tff(fact_2502_card__Diff1__less,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)))),finite_card_o(A2)) ) ) ).

% card_Diff1_less
tff(fact_2503_card__Diff1__less,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)))),finite_card_set_nat(A2)) ) ) ).

% card_Diff1_less
tff(fact_2504_card__Diff1__less,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))),finite_card_real(A2)) ) ) ).

% card_Diff1_less
tff(fact_2505_card__Diff1__less,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A2)) ) ) ).

% card_Diff1_less
tff(fact_2506_card__Diff1__less,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)))),finite_card_nat(A2)) ) ) ).

% card_Diff1_less
tff(fact_2507_Inf__fin_Oinsert,axiom,
    ! [A2: set_se7855581050983116737at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( A2 != bot_bo3083307316010499117at_nat )
       => ( lattic30941717366863870at_nat(insert9200635055090092081at_nat(Xa,A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),lattic30941717366863870at_nat(A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2508_Inf__fin_Oinsert,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( lattic5238388535129920115in_nat(insert_nat(Xa,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),lattic5238388535129920115in_nat(A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2509_Inf__fin_Oinsert,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( lattic5235898064620869839in_int(insert_int(Xa,A2)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),lattic5235898064620869839in_int(A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2510_Inf__fin_Oinsert,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((Xa),A2))
        <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2511_Inf__fin_Oinsert,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( lattic3014633134055518761et_nat(insert_set_nat(Xa,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),lattic3014633134055518761et_nat(A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2512_Inf__fin_Oinsert,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( lattic2677971596711400399n_real(insert_real(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,Xa),lattic2677971596711400399n_real(A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2513_Inf__fin_Oinsert,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( lattic974744108425517955d_enat(insert_Extended_enat(Xa,A2)) = inf_in1870772243966228564d_enat(Xa,lattic974744108425517955d_enat(A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_2514_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_2515_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_2516_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_2517_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_2518_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( aa_set_nat_o(finite_finite_nat,F2)
        | aa_set_nat_o(finite_finite_nat,G) )
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,F2),G)) ) ).

% finite_Int
tff(fact_2519_finite__Int,axiom,
    ! [F2: set_int,G: set_int] :
      ( ( aa_set_int_o(finite_finite_int,F2)
        | aa_set_int_o(finite_finite_int,G) )
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,F2),G)) ) ).

% finite_Int
tff(fact_2520_finite__Int,axiom,
    ! [F2: set_complex,G: set_complex] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,F2)
        | aa_set_complex_o(finite3207457112153483333omplex,G) )
     => aa_set_complex_o(finite3207457112153483333omplex,inf_inf_set_complex(F2,G)) ) ).

% finite_Int
tff(fact_2521_finite__Int,axiom,
    ! [F2: set_Extended_enat,G: set_Extended_enat] :
      ( ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
        | aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,G) )
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,inf_in8357106775501769908d_enat(F2,G)) ) ).

% finite_Int
tff(fact_2522_finite__Int,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat(F2)
        | finite6177210948735845034at_nat(G) )
     => finite6177210948735845034at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,F2),G)) ) ).

% finite_Int
tff(fact_2523_diff__0__eq__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Na) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_2524_diff__self__eq__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),M2) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_2525_Int__subset__iff,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))
    <=> ( ord_le3146513528884898305at_nat(C,A2)
        & ord_le3146513528884898305at_nat(C,B) ) ) ).

% Int_subset_iff
tff(fact_2526_Int__subset__iff,axiom,
    ! [C: set_int,A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))
    <=> ( aa_set_int_o(ord_less_eq_set_int(C),A2)
        & aa_set_int_o(ord_less_eq_set_int(C),B) ) ) ).

% Int_subset_iff
tff(fact_2527_inf__right__idem,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) ).

% inf_right_idem
tff(fact_2528_inf__right__idem,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),Ya) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) ).

% inf_right_idem
tff(fact_2529_inf_Oright__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) ).

% inf.right_idem
tff(fact_2530_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2)),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) ).

% inf.right_idem
tff(fact_2531_inf__left__idem,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) ).

% inf_left_idem
tff(fact_2532_inf__left__idem,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) ).

% inf_left_idem
tff(fact_2533_inf_Oleft__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) ).

% inf.left_idem
tff(fact_2534_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) ).

% inf.left_idem
tff(fact_2535_inf__idem,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Xa) = Xa ).

% inf_idem
tff(fact_2536_inf__idem,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Xa) = Xa ).

% inf_idem
tff(fact_2537_inf_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),A) = A ).

% inf.idem
tff(fact_2538_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),A) = A ).

% inf.idem
tff(fact_2539_Int__insert__right__if1,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ( inf_in8357106775501769908d_enat(A2,insert_Extended_enat(A,B)) = insert_Extended_enat(A,inf_in8357106775501769908d_enat(A2,B)) ) ) ).

% Int_insert_right_if1
tff(fact_2540_Int__insert__right__if1,axiom,
    ! [A: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( inf_inf_set_real(A2,insert_real(A,B)) = insert_real(A,inf_inf_set_real(A2,B)) ) ) ).

% Int_insert_right_if1
tff(fact_2541_Int__insert__right__if1,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( inf_inf_set_set_nat(A2,insert_set_nat(A,B)) = insert_set_nat(A,inf_inf_set_set_nat(A2,B)) ) ) ).

% Int_insert_right_if1
tff(fact_2542_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(A,B)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2543_Int__insert__right__if1,axiom,
    ! [A: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(A,B)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2544_Int__insert__right__if1,axiom,
    ! [A: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((A),B)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2545_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(A,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2546_Int__insert__right__if0,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ( inf_in8357106775501769908d_enat(A2,insert_Extended_enat(A,B)) = inf_in8357106775501769908d_enat(A2,B) ) ) ).

% Int_insert_right_if0
tff(fact_2547_Int__insert__right__if0,axiom,
    ! [A: real,A2: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( inf_inf_set_real(A2,insert_real(A,B)) = inf_inf_set_real(A2,B) ) ) ).

% Int_insert_right_if0
tff(fact_2548_Int__insert__right__if0,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A2)
     => ( inf_inf_set_set_nat(A2,insert_set_nat(A,B)) = inf_inf_set_set_nat(A2,B) ) ) ).

% Int_insert_right_if0
tff(fact_2549_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(A,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) ) ) ).

% Int_insert_right_if0
tff(fact_2550_Int__insert__right__if0,axiom,
    ! [A: int,A2: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(A,B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) ) ) ).

% Int_insert_right_if0
tff(fact_2551_Int__insert__right__if0,axiom,
    ! [A: $o,A2: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((A),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) ) ) ).

% Int_insert_right_if0
tff(fact_2552_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ member8440522571783428010at_nat(A,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) ) ) ).

% Int_insert_right_if0
tff(fact_2553_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A2)),insert_nat(A,B)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)) ).

% insert_inter_insert
tff(fact_2554_insert__inter__insert,axiom,
    ! [A: int,A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A2)),insert_int(A,B)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) ).

% insert_inter_insert
tff(fact_2555_insert__inter__insert,axiom,
    ! [A: $o,A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),A2)),insert_o((A),B)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)) ).

% insert_inter_insert
tff(fact_2556_insert__inter__insert,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_set_nat] : inf_inf_set_set_nat(insert_set_nat(A,A2),insert_set_nat(A,B)) = insert_set_nat(A,inf_inf_set_set_nat(A2,B)) ).

% insert_inter_insert
tff(fact_2557_insert__inter__insert,axiom,
    ! [A: real,A2: set_real,B: set_real] : inf_inf_set_real(insert_real(A,A2),insert_real(A,B)) = insert_real(A,inf_inf_set_real(A2,B)) ).

% insert_inter_insert
tff(fact_2558_insert__inter__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] : inf_in8357106775501769908d_enat(insert_Extended_enat(A,A2),insert_Extended_enat(A,B)) = insert_Extended_enat(A,inf_in8357106775501769908d_enat(A2,B)) ).

% insert_inter_insert
tff(fact_2559_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),insert8211810215607154385at_nat(A,B)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) ).

% insert_inter_insert
tff(fact_2560_Int__insert__left__if1,axiom,
    ! [A: extended_enat,C: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),C)
     => ( inf_in8357106775501769908d_enat(insert_Extended_enat(A,B),C) = insert_Extended_enat(A,inf_in8357106775501769908d_enat(B,C)) ) ) ).

% Int_insert_left_if1
tff(fact_2561_Int__insert__left__if1,axiom,
    ! [A: real,C: set_real,B: set_real] :
      ( aa_set_real_o(member_real(A),C)
     => ( inf_inf_set_real(insert_real(A,B),C) = insert_real(A,inf_inf_set_real(B,C)) ) ) ).

% Int_insert_left_if1
tff(fact_2562_Int__insert__left__if1,axiom,
    ! [A: set_nat,C: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),C)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B),C) = insert_set_nat(A,inf_inf_set_set_nat(B,C)) ) ) ).

% Int_insert_left_if1
tff(fact_2563_Int__insert__left__if1,axiom,
    ! [A: nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(A),C)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B)),C) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ) ).

% Int_insert_left_if1
tff(fact_2564_Int__insert__left__if1,axiom,
    ! [A: int,C: set_int,B: set_int] :
      ( aa_set_int_o(member_int(A),C)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B)),C) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C)) ) ) ).

% Int_insert_left_if1
tff(fact_2565_Int__insert__left__if1,axiom,
    ! [A: $o,C: set_o,B: set_o] :
      ( aa_set_o_o(member_o((A)),C)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),B)),C) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C)) ) ) ).

% Int_insert_left_if1
tff(fact_2566_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(A,C)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B)),C) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ) ) ).

% Int_insert_left_if1
tff(fact_2567_Int__insert__left__if0,axiom,
    ! [A: extended_enat,C: set_Extended_enat,B: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),C)
     => ( inf_in8357106775501769908d_enat(insert_Extended_enat(A,B),C) = inf_in8357106775501769908d_enat(B,C) ) ) ).

% Int_insert_left_if0
tff(fact_2568_Int__insert__left__if0,axiom,
    ! [A: real,C: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(A),C)
     => ( inf_inf_set_real(insert_real(A,B),C) = inf_inf_set_real(B,C) ) ) ).

% Int_insert_left_if0
tff(fact_2569_Int__insert__left__if0,axiom,
    ! [A: set_nat,C: set_set_nat,B: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),C)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B),C) = inf_inf_set_set_nat(B,C) ) ) ).

% Int_insert_left_if0
tff(fact_2570_Int__insert__left__if0,axiom,
    ! [A: nat,C: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),C)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C) ) ) ).

% Int_insert_left_if0
tff(fact_2571_Int__insert__left__if0,axiom,
    ! [A: int,C: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(A),C)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B)),C) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C) ) ) ).

% Int_insert_left_if0
tff(fact_2572_Int__insert__left__if0,axiom,
    ! [A: $o,C: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((A)),C)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),B)),C) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C) ) ) ).

% Int_insert_left_if0
tff(fact_2573_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ member8440522571783428010at_nat(A,C)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C) ) ) ).

% Int_insert_left_if0
tff(fact_2574_of__nat__eq__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M2 = Na ) ) ).

% of_nat_eq_iff
tff(fact_2575_of__nat__eq__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( M2 = Na ) ) ).

% of_nat_eq_iff
tff(fact_2576_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_2577_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_2578_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_2579_Un__Int__eq_I1_J,axiom,
    ! [S: set_o,T2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_2580_Un__Int__eq_I1_J,axiom,
    ! [S: set_int,T2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_2581_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_2582_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_2583_Un__Int__eq_I2_J,axiom,
    ! [S: set_o,T2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_2584_Un__Int__eq_I2_J,axiom,
    ! [S: set_int,T2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_2585_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_2586_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_2587_Un__Int__eq_I3_J,axiom,
    ! [S: set_o,T2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,S),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_2588_Un__Int__eq_I3_J,axiom,
    ! [S: set_int,T2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_2589_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_2590_Un__Int__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_2591_Un__Int__eq_I4_J,axiom,
    ! [T2: set_o,S: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,T2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_2592_Un__Int__eq_I4_J,axiom,
    ! [T2: set_int,S: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,T2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_2593_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_2594_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_2595_Int__Un__eq_I1_J,axiom,
    ! [S: set_o,T2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_2596_Int__Un__eq_I1_J,axiom,
    ! [S: set_int,T2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_2597_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_2598_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_2599_Int__Un__eq_I2_J,axiom,
    ! [S: set_o,T2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_2600_Int__Un__eq_I2_J,axiom,
    ! [S: set_int,T2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_2601_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_2602_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_2603_Int__Un__eq_I3_J,axiom,
    ! [S: set_o,T2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_2604_Int__Un__eq_I3_J,axiom,
    ! [S: set_int,T2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_2605_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_2606_Int__Un__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_2607_Int__Un__eq_I4_J,axiom,
    ! [T2: set_o,S: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,T2),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_2608_Int__Un__eq_I4_J,axiom,
    ! [T2: set_int,S: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,T2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_2609_member__remove,axiom,
    ! [Xa: extended_enat,Ya: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),remove_Extended_enat(Ya,A2))
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
        & ( Xa != Ya ) ) ) ).

% member_remove
tff(fact_2610_member__remove,axiom,
    ! [Xa: real,Ya: real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),remove_real(Ya,A2))
    <=> ( aa_set_real_o(member_real(Xa),A2)
        & ( Xa != Ya ) ) ) ).

% member_remove
tff(fact_2611_member__remove,axiom,
    ! [Xa: set_nat,Ya: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),remove_set_nat(Ya,A2))
    <=> ( aa_set_set_nat_o(member_set_nat(Xa),A2)
        & ( Xa != Ya ) ) ) ).

% member_remove
tff(fact_2612_member__remove,axiom,
    ! [Xa: nat,Ya: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),remove_nat(Ya,A2))
    <=> ( aa_set_nat_o(member_nat(Xa),A2)
        & ( Xa != Ya ) ) ) ).

% member_remove
tff(fact_2613_member__remove,axiom,
    ! [Xa: int,Ya: int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),remove_int(Ya,A2))
    <=> ( aa_set_int_o(member_int(Xa),A2)
        & ( Xa != Ya ) ) ) ).

% member_remove
tff(fact_2614_member__remove,axiom,
    ! [Xa: $o,Ya: $o,A2: set_o] :
      ( aa_set_o_o(member_o((Xa)),remove_o((Ya),A2))
    <=> ( aa_set_o_o(member_o((Xa)),A2)
        & ~ ( (Xa)
          <=> (Ya) ) ) ) ).

% member_remove
tff(fact_2615_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2))
    <=> ( ord_le3146513528884898305at_nat(A,B2)
        & ord_le3146513528884898305at_nat(A,C2) ) ) ).

% inf.bounded_iff
tff(fact_2616_inf_Obounded__iff,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B2)
        & aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2617_inf_Obounded__iff,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B2),C2))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2618_inf_Obounded__iff,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2619_inf_Obounded__iff,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2620_le__inf__iff,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z))
    <=> ( ord_le3146513528884898305at_nat(Xa,Ya)
        & ord_le3146513528884898305at_nat(Xa,Z) ) ) ).

% le_inf_iff
tff(fact_2621_le__inf__iff,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z))
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
        & aa_set_int_o(ord_less_eq_set_int(Xa),Z) ) ) ).

% le_inf_iff
tff(fact_2622_le__inf__iff,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Ya),Z))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) ) ) ).

% le_inf_iff
tff(fact_2623_le__inf__iff,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) ) ) ).

% le_inf_iff
tff(fact_2624_le__inf__iff,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Ya),Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) ) ) ).

% le_inf_iff
tff(fact_2625_zero__less__diff,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% zero_less_diff
tff(fact_2626_inf__bot__right,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% inf_bot_right
tff(fact_2627_inf__bot__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_2628_inf__bot__right,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_2629_inf__bot__right,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),bot_bot_set_o) = bot_bot_set_o ).

% inf_bot_right
tff(fact_2630_inf__bot__right,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),bot_bot_filter_nat) = bot_bot_filter_nat ).

% inf_bot_right
tff(fact_2631_inf__bot__right,axiom,
    ! [Xa: set_set_nat] : inf_inf_set_set_nat(Xa,bot_bot_set_set_nat) = bot_bot_set_set_nat ).

% inf_bot_right
tff(fact_2632_inf__bot__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,bot_bot_set_real) = bot_bot_set_real ).

% inf_bot_right
tff(fact_2633_inf__bot__right,axiom,
    ! [Xa: set_Extended_enat] : inf_in8357106775501769908d_enat(Xa,bot_bo7653980558646680370d_enat) = bot_bo7653980558646680370d_enat ).

% inf_bot_right
tff(fact_2634_inf__bot__left,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),Xa) = bot_bo2099793752762293965at_nat ).

% inf_bot_left
tff(fact_2635_inf__bot__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),Xa) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_2636_inf__bot__left,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),Xa) = bot_bot_set_int ).

% inf_bot_left
tff(fact_2637_inf__bot__left,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),Xa) = bot_bot_set_o ).

% inf_bot_left
tff(fact_2638_inf__bot__left,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,bot_bot_filter_nat),Xa) = bot_bot_filter_nat ).

% inf_bot_left
tff(fact_2639_inf__bot__left,axiom,
    ! [Xa: set_set_nat] : inf_inf_set_set_nat(bot_bot_set_set_nat,Xa) = bot_bot_set_set_nat ).

% inf_bot_left
tff(fact_2640_inf__bot__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(bot_bot_set_real,Xa) = bot_bot_set_real ).

% inf_bot_left
tff(fact_2641_inf__bot__left,axiom,
    ! [Xa: set_Extended_enat] : inf_in8357106775501769908d_enat(bot_bo7653980558646680370d_enat,Xa) = bot_bo7653980558646680370d_enat ).

% inf_bot_left
tff(fact_2642_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_2643_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_2644_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_2645_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),bot_bot_set_o) = bot_bot_set_o ).

% boolean_algebra.conj_zero_right
tff(fact_2646_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_set_nat] : inf_inf_set_set_nat(Xa,bot_bot_set_set_nat) = bot_bot_set_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_2647_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,bot_bot_set_real) = bot_bot_set_real ).

% boolean_algebra.conj_zero_right
tff(fact_2648_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_Extended_enat] : inf_in8357106775501769908d_enat(Xa,bot_bo7653980558646680370d_enat) = bot_bo7653980558646680370d_enat ).

% boolean_algebra.conj_zero_right
tff(fact_2649_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),Xa) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_2650_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),Xa) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_2651_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),Xa) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_2652_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),Xa) = bot_bot_set_o ).

% boolean_algebra.conj_zero_left
tff(fact_2653_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_set_nat] : inf_inf_set_set_nat(bot_bot_set_set_nat,Xa) = bot_bot_set_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_2654_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(bot_bot_set_real,Xa) = bot_bot_set_real ).

% boolean_algebra.conj_zero_left
tff(fact_2655_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_Extended_enat] : inf_in8357106775501769908d_enat(bot_bo7653980558646680370d_enat,Xa) = bot_bo7653980558646680370d_enat ).

% boolean_algebra.conj_zero_left
tff(fact_2656_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B) = bot_bo2099793752762293965at_nat )
    <=> ( ~ member8440522571783428010at_nat(A,B)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_2657_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A2)),B) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_2658_insert__disjoint_I1_J,axiom,
    ! [A: int,A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A2)),B) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_2659_insert__disjoint_I1_J,axiom,
    ! [A: $o,A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),A2)),B) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B)
        & ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_2660_insert__disjoint_I1_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( inf_inf_set_set_nat(insert_set_nat(A,A2),B) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B)
        & ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_2661_insert__disjoint_I1_J,axiom,
    ! [A: real,A2: set_real,B: set_real] :
      ( ( inf_inf_set_real(insert_real(A,A2),B) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B)
        & ( inf_inf_set_real(A2,B) = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
tff(fact_2662_insert__disjoint_I1_J,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(insert_Extended_enat(A,A2),B) = bot_bo7653980558646680370d_enat )
    <=> ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),B)
        & ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat ) ) ) ).

% insert_disjoint(1)
tff(fact_2663_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B) )
    <=> ( ~ member8440522571783428010at_nat(A,B)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2664_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A2)),B) )
    <=> ( ~ aa_set_nat_o(member_nat(A),B)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2665_insert__disjoint_I2_J,axiom,
    ! [A: int,A2: set_int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A2)),B) )
    <=> ( ~ aa_set_int_o(member_int(A),B)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2666_insert__disjoint_I2_J,axiom,
    ! [A: $o,A2: set_o,B: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),A2)),B) )
    <=> ( ~ aa_set_o_o(member_o((A)),B)
        & ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2667_insert__disjoint_I2_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(insert_set_nat(A,A2),B) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2668_insert__disjoint_I2_J,axiom,
    ! [A: real,A2: set_real,B: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(insert_real(A,A2),B) )
    <=> ( ~ aa_set_real_o(member_real(A),B)
        & ( bot_bot_set_real = inf_inf_set_real(A2,B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2669_insert__disjoint_I2_J,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = inf_in8357106775501769908d_enat(insert_Extended_enat(A,A2),B) )
    <=> ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),B)
        & ( bot_bo7653980558646680370d_enat = inf_in8357106775501769908d_enat(A2,B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2670_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),insert8211810215607154385at_nat(A,A2)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ member8440522571783428010at_nat(A,B)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A2) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_2671_disjoint__insert_I1_J,axiom,
    ! [B: set_nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),insert_nat(A,A2)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A2) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_2672_disjoint__insert_I1_J,axiom,
    ! [B: set_int,A: int,A2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),insert_int(A,A2)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),A2) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_2673_disjoint__insert_I1_J,axiom,
    ! [B: set_o,A: $o,A2: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),insert_o((A),A2)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B)
        & ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),A2) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_2674_disjoint__insert_I1_J,axiom,
    ! [B: set_set_nat,A: set_nat,A2: set_set_nat] :
      ( ( inf_inf_set_set_nat(B,insert_set_nat(A,A2)) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B)
        & ( inf_inf_set_set_nat(B,A2) = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_2675_disjoint__insert_I1_J,axiom,
    ! [B: set_real,A: real,A2: set_real] :
      ( ( inf_inf_set_real(B,insert_real(A,A2)) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B)
        & ( inf_inf_set_real(B,A2) = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
tff(fact_2676_disjoint__insert_I1_J,axiom,
    ! [B: set_Extended_enat,A: extended_enat,A2: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(B,insert_Extended_enat(A,A2)) = bot_bo7653980558646680370d_enat )
    <=> ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),B)
        & ( inf_in8357106775501769908d_enat(B,A2) = bot_bo7653980558646680370d_enat ) ) ) ).

% disjoint_insert(1)
tff(fact_2677_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(B2,B)) )
    <=> ( ~ member8440522571783428010at_nat(B2,A2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2678_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B2: nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(B2,B)) )
    <=> ( ~ aa_set_nat_o(member_nat(B2),A2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2679_disjoint__insert_I2_J,axiom,
    ! [A2: set_int,B2: int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(B2,B)) )
    <=> ( ~ aa_set_int_o(member_int(B2),A2)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2680_disjoint__insert_I2_J,axiom,
    ! [A2: set_o,B2: $o,B: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((B2),B)) )
    <=> ( ~ aa_set_o_o(member_o((B2)),A2)
        & ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2681_disjoint__insert_I2_J,axiom,
    ! [A2: set_set_nat,B2: set_nat,B: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,insert_set_nat(B2,B)) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(B2),A2)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2682_disjoint__insert_I2_J,axiom,
    ! [A2: set_real,B2: real,B: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(A2,insert_real(B2,B)) )
    <=> ( ~ aa_set_real_o(member_real(B2),A2)
        & ( bot_bot_set_real = inf_inf_set_real(A2,B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2683_disjoint__insert_I2_J,axiom,
    ! [A2: set_Extended_enat,B2: extended_enat,B: set_Extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = inf_in8357106775501769908d_enat(A2,insert_Extended_enat(B2,B)) )
    <=> ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(B2),A2)
        & ( bot_bo7653980558646680370d_enat = inf_in8357106775501769908d_enat(A2,B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2684_diff__is__0__eq_H,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2685_diff__is__0__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% diff_is_0_eq
tff(fact_2686_sup__inf__absorb,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)) = Xa ).

% sup_inf_absorb
tff(fact_2687_sup__inf__absorb,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)) = Xa ).

% sup_inf_absorb
tff(fact_2688_sup__inf__absorb,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)) = Xa ).

% sup_inf_absorb
tff(fact_2689_sup__inf__absorb,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)) = Xa ).

% sup_inf_absorb
tff(fact_2690_sup__inf__absorb,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)) = Xa ).

% sup_inf_absorb
tff(fact_2691_sup__inf__absorb,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Ya)) = Xa ).

% sup_inf_absorb
tff(fact_2692_inf__sup__absorb,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Ya)) = Xa ).

% inf_sup_absorb
tff(fact_2693_inf__sup__absorb,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) = Xa ).

% inf_sup_absorb
tff(fact_2694_inf__sup__absorb,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)) = Xa ).

% inf_sup_absorb
tff(fact_2695_inf__sup__absorb,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) = Xa ).

% inf_sup_absorb
tff(fact_2696_inf__sup__absorb,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) = Xa ).

% inf_sup_absorb
tff(fact_2697_inf__sup__absorb,axiom,
    ! [Xa: filter_nat,Ya: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)) = Xa ).

% inf_sup_absorb
tff(fact_2698_Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),A2)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_2699_Diff__disjoint,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_2700_Diff__disjoint,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,B),A2)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_2701_Diff__disjoint,axiom,
    ! [A2: set_set_nat,B: set_set_nat] : inf_inf_set_set_nat(A2,minus_2163939370556025621et_nat(B,A2)) = bot_bot_set_set_nat ).

% Diff_disjoint
tff(fact_2702_Diff__disjoint,axiom,
    ! [A2: set_real,B: set_real] : inf_inf_set_real(A2,minus_minus_set_real(B,A2)) = bot_bot_set_real ).

% Diff_disjoint
tff(fact_2703_Diff__disjoint,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] : inf_in8357106775501769908d_enat(A2,minus_925952699566721837d_enat(B,A2)) = bot_bo7653980558646680370d_enat ).

% Diff_disjoint
tff(fact_2704_Diff__disjoint,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_2705_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_2706_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_2707_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_2708_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_2709_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_2710_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_2711_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_2712_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_2713_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M2) = zero_zero_rat )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_2714_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M2) = zero_zero_nat )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_2715_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = zero_zero_int )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_2716_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = zero_zero_real )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_2717_of__nat__less__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_iff
tff(fact_2718_of__nat__less__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_iff
tff(fact_2719_of__nat__less__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_iff
tff(fact_2720_of__nat__less__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_iff
tff(fact_2721_of__nat__le__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% of_nat_le_iff
tff(fact_2722_of__nat__le__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% of_nat_le_iff
tff(fact_2723_of__nat__le__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% of_nat_le_iff
tff(fact_2724_of__nat__le__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% of_nat_le_iff
tff(fact_2725_card_Oempty,axiom,
    finite_card_complex(bot_bot_set_complex) = zero_zero_nat ).

% card.empty
tff(fact_2726_card_Oempty,axiom,
    finite410649719033368117t_unit(bot_bo3957492148770167129t_unit) = zero_zero_nat ).

% card.empty
tff(fact_2727_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_2728_card_Oempty,axiom,
    finite_card_nat(bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_2729_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_2730_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_2731_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_2732_card_Oempty,axiom,
    finite_card_real(bot_bot_set_real) = zero_zero_nat ).

% card.empty
tff(fact_2733_card_Oempty,axiom,
    finite121521170596916366d_enat(bot_bo7653980558646680370d_enat) = zero_zero_nat ).

% card.empty
tff(fact_2734_card_Oinfinite,axiom,
    ! [A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite_card_set_nat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2735_card_Oinfinite,axiom,
    ! [A2: set_Product_unit] :
      ( ~ finite4290736615968046902t_unit(A2)
     => ( finite410649719033368117t_unit(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2736_card_Oinfinite,axiom,
    ! [A2: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(A2)
     => ( finite_card_list_nat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2737_card_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_nat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2738_card_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_int(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2739_card_Oinfinite,axiom,
    ! [A2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite_card_complex(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2740_card_Oinfinite,axiom,
    ! [A2: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite121521170596916366d_enat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2741_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Na) = one_one_complex )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2742_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2743_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2744_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2745_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_2746_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2747_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2748_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2749_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2750_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_2751_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_2752_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_2753_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_2754_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_2755_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_2756_inf__Sup__absorb,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),lattic8928443293348198069n_real(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_2757_inf__Sup__absorb,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),lattic3835124923745554447et_nat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_2758_inf__Sup__absorb,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((A)),A2)
       => ( aa_o_o(aa_o_fun_o_o(inf_inf_o,(A)),aa_set_o_o(lattic1508158080041050831_fin_o,A2))
        <=> (A) ) ) ) ).

% inf_Sup_absorb
tff(fact_2759_inf__Sup__absorb,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),lattic1091506334969745077in_int(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_2760_inf__Sup__absorb,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => ( inf_in1870772243966228564d_enat(A,lattic5005175426920976669d_enat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_2761_inf__Sup__absorb,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),lattic1093996805478795353in_nat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_2762_inf__Sup__absorb,axiom,
    ! [A2: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( member2643936169264416010at_nat(A,A2)
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),lattic1541023418247406232at_nat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_2763_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),zero_zero_real)
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2764_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),zero_zero_rat)
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2765_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),zero_zero_nat)
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2766_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),zero_zero_int)
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_2767_card__0__eq,axiom,
    ! [A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ( finite410649719033368117t_unit(A2) = zero_zero_nat )
      <=> ( A2 = bot_bo3957492148770167129t_unit ) ) ) ).

% card_0_eq
tff(fact_2768_card__0__eq,axiom,
    ! [A2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( ( finite_card_list_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_2769_card__0__eq,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( finite_card_complex(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_complex ) ) ) ).

% card_0_eq
tff(fact_2770_card__0__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( finite_card_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_2771_card__0__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( finite_card_int(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_2772_card__0__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( finite_card_o(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_2773_card__0__eq,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( finite_card_set_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_2774_card__0__eq,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( finite_card_real(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_real ) ) ) ).

% card_0_eq
tff(fact_2775_card__0__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( finite121521170596916366d_enat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ) ).

% card_0_eq
tff(fact_2776_card__Diff__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A),B)
       => ( finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2777_card__Diff__insert,axiom,
    ! [A: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( ~ aa_set_real_o(member_real(A),B)
       => ( finite_card_real(minus_minus_set_real(A2,insert_real(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(minus_minus_set_real(A2,B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2778_card__Diff__insert,axiom,
    ! [A: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( ~ aa_set_int_o(member_int(A),B)
       => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2779_card__Diff__insert,axiom,
    ! [A: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( ~ aa_set_o_o(member_o((A)),B)
       => ( finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2780_card__Diff__insert,axiom,
    ! [A: complex,A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(member_complex(A),A2)
     => ( ~ aa_set_complex_o(member_complex(A),B)
       => ( finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(minus_811609699411566653omplex(A2,B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2781_card__Diff__insert,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(A),B)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2782_card__Diff__insert,axiom,
    ! [A: product_unit,A2: set_Product_unit,B: set_Product_unit] :
      ( member_Product_unit(A,A2)
     => ( ~ member_Product_unit(A,B)
       => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2783_card__Diff__insert,axiom,
    ! [A: list_nat,A2: set_list_nat,B: set_list_nat] :
      ( member_list_nat(A,A2)
     => ( ~ member_list_nat(A,B)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2784_card__Diff__insert,axiom,
    ! [A: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( ~ aa_set_nat_o(member_nat(A),B)
       => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(A,B))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2785_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: nat,K: nat,B2: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_2786_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),B2) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_2787_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_2788_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),A) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_2789_inf__left__commute,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ).

% inf_left_commute
tff(fact_2790_inf__left__commute,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Z)) ).

% inf_left_commute
tff(fact_2791_inf_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C2)) ).

% inf.left_commute
tff(fact_2792_inf_Oleft__commute,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% inf.left_commute
tff(fact_2793_inf__commute,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Xa) ).

% inf_commute
tff(fact_2794_inf__commute,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Xa) ).

% inf_commute
tff(fact_2795_inf_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),A) ).

% inf.commute
tff(fact_2796_inf_Ocommute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A) ).

% inf.commute
tff(fact_2797_inf__assoc,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) ).

% inf_assoc
tff(fact_2798_inf__assoc,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) ).

% inf_assoc
tff(fact_2799_inf_Oassoc,axiom,
    ! [A: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C2)) ).

% inf.assoc
tff(fact_2800_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% inf.assoc
tff(fact_2801_inf__sup__aci_I1_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Xa) ).

% inf_sup_aci(1)
tff(fact_2802_inf__sup__aci_I1_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Xa) ).

% inf_sup_aci(1)
tff(fact_2803_inf__sup__aci_I2_J,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) ).

% inf_sup_aci(2)
tff(fact_2804_inf__sup__aci_I2_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) ).

% inf_sup_aci(2)
tff(fact_2805_inf__sup__aci_I3_J,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_2806_inf__sup__aci_I3_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Z)) ).

% inf_sup_aci(3)
tff(fact_2807_inf__sup__aci_I4_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) ).

% inf_sup_aci(4)
tff(fact_2808_inf__sup__aci_I4_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) ).

% inf_sup_aci(4)
tff(fact_2809_card__Diff__subset__Int,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,inf_inf_set_set_nat(A2,B))
     => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),finite_card_set_nat(inf_inf_set_set_nat(A2,B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2810_card__Diff__subset__Int,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( finite4290736615968046902t_unit(inf_in4660618365625256667t_unit(A2,B))
     => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(inf_in4660618365625256667t_unit(A2,B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2811_card__Diff__subset__Int,axiom,
    ! [A2: set_list_nat,B: set_list_nat] :
      ( finite8100373058378681591st_nat(inf_inf_set_list_nat(A2,B))
     => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),finite_card_list_nat(inf_inf_set_list_nat(A2,B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2812_card__Diff__subset__Int,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))
     => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2813_card__Diff__subset__Int,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,inf_inf_set_complex(A2,B))
     => ( finite_card_complex(minus_811609699411566653omplex(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),finite_card_complex(inf_inf_set_complex(A2,B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2814_card__Diff__subset__Int,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,inf_in8357106775501769908d_enat(A2,B))
     => ( finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(inf_in8357106775501769908d_enat(A2,B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2815_card__Diff__subset__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))
     => ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite711546835091564841at_nat(A2)),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2816_card__Diff__subset__Int,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))
     => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))) ) ) ).

% card_Diff_subset_Int
tff(fact_2817_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xa: nat,Ya: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Ya)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Ya))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_2818_inf_OcoboundedI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B2,C2)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),C2) ) ).

% inf.coboundedI2
tff(fact_2819_inf_OcoboundedI2,axiom,
    ! [B2: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),C2)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),C2) ) ).

% inf.coboundedI2
tff(fact_2820_inf_OcoboundedI2,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( 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,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),C2) ) ).

% inf.coboundedI2
tff(fact_2821_inf_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( 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(inf_inf_nat,A),B2)),C2) ) ).

% inf.coboundedI2
tff(fact_2822_inf_OcoboundedI2,axiom,
    ! [B2: int,C2: int,A: int] :
      ( 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,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),C2) ) ).

% inf.coboundedI2
tff(fact_2823_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,C2)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),C2) ) ).

% inf.coboundedI1
tff(fact_2824_inf_OcoboundedI1,axiom,
    ! [A: set_int,C2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),C2)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),C2) ) ).

% inf.coboundedI1
tff(fact_2825_inf_OcoboundedI1,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),C2) ) ).

% inf.coboundedI1
tff(fact_2826_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),C2) ) ).

% inf.coboundedI1
tff(fact_2827_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),C2) ) ).

% inf.coboundedI1
tff(fact_2828_inf_Oabsorb__iff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B2,A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_2829_inf_Oabsorb__iff2,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_2830_inf_Oabsorb__iff2,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_2831_inf_Oabsorb__iff2,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_2832_inf_Oabsorb__iff2,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_2833_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B2)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = A ) ) ).

% inf.absorb_iff1
tff(fact_2834_inf_Oabsorb__iff1,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) = A ) ) ).

% inf.absorb_iff1
tff(fact_2835_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) = A ) ) ).

% inf.absorb_iff1
tff(fact_2836_inf_Oabsorb__iff1,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = A ) ) ).

% inf.absorb_iff1
tff(fact_2837_inf_Oabsorb__iff1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = A ) ) ).

% inf.absorb_iff1
tff(fact_2838_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),B2) ).

% inf.cobounded2
tff(fact_2839_inf_Ocobounded2,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),B2) ).

% inf.cobounded2
tff(fact_2840_inf_Ocobounded2,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),B2) ).

% inf.cobounded2
tff(fact_2841_inf_Ocobounded2,axiom,
    ! [A: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),B2) ).

% inf.cobounded2
tff(fact_2842_inf_Ocobounded2,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),B2) ).

% inf.cobounded2
tff(fact_2843_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),A) ).

% inf.cobounded1
tff(fact_2844_inf_Ocobounded1,axiom,
    ! [A: set_int,B2: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),A) ).

% inf.cobounded1
tff(fact_2845_inf_Ocobounded1,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),A) ).

% inf.cobounded1
tff(fact_2846_inf_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),A) ).

% inf.cobounded1
tff(fact_2847_inf_Ocobounded1,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),A) ).

% inf.cobounded1
tff(fact_2848_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B2)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) ) ) ).

% inf.order_iff
tff(fact_2849_inf_Oorder__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) ) ) ).

% inf.order_iff
tff(fact_2850_inf_Oorder__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) ) ) ).

% inf.order_iff
tff(fact_2851_inf_Oorder__iff,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) ) ) ).

% inf.order_iff
tff(fact_2852_inf_Oorder__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) ) ) ).

% inf.order_iff
tff(fact_2853_inf__greatest,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Xa,Ya)
     => ( ord_le3146513528884898305at_nat(Xa,Z)
       => ord_le3146513528884898305at_nat(Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) ) ) ).

% inf_greatest
tff(fact_2854_inf__greatest,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_o(ord_less_eq_set_int(Xa),Z)
       => aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z)) ) ) ).

% inf_greatest
tff(fact_2855_inf__greatest,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Ya),Z)) ) ) ).

% inf_greatest
tff(fact_2856_inf__greatest,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) ) ) ).

% inf_greatest
tff(fact_2857_inf__greatest,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Ya),Z)) ) ) ).

% inf_greatest
tff(fact_2858_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B2)
     => ( ord_le3146513528884898305at_nat(A,C2)
       => ord_le3146513528884898305at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ) ).

% inf.boundedI
tff(fact_2859_inf_OboundedI,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ) ) ).

% inf.boundedI
tff(fact_2860_inf_OboundedI,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B2),C2)) ) ) ).

% inf.boundedI
tff(fact_2861_inf_OboundedI,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C2)) ) ) ).

% inf.boundedI
tff(fact_2862_inf_OboundedI,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C2)) ) ) ).

% inf.boundedI
tff(fact_2863_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2))
     => ~ ( ord_le3146513528884898305at_nat(A,B2)
         => ~ ord_le3146513528884898305at_nat(A,C2) ) ) ).

% inf.boundedE
tff(fact_2864_inf_OboundedE,axiom,
    ! [A: set_int,B2: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2))
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A),B2)
         => ~ aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% inf.boundedE
tff(fact_2865_inf_OboundedE,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B2),C2))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% inf.boundedE
tff(fact_2866_inf_OboundedE,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% inf.boundedE
tff(fact_2867_inf_OboundedE,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% inf.boundedE
tff(fact_2868_inf__absorb2,axiom,
    ! [Ya: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Ya,Xa)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = Ya ) ) ).

% inf_absorb2
tff(fact_2869_inf__absorb2,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya) = Ya ) ) ).

% inf_absorb2
tff(fact_2870_inf__absorb2,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya) = Ya ) ) ).

% inf_absorb2
tff(fact_2871_inf__absorb2,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) = Ya ) ) ).

% inf_absorb2
tff(fact_2872_inf__absorb2,axiom,
    ! [Ya: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya) = Ya ) ) ).

% inf_absorb2
tff(fact_2873_inf__absorb1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Xa,Ya)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = Xa ) ) ).

% inf_absorb1
tff(fact_2874_inf__absorb1,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya) = Xa ) ) ).

% inf_absorb1
tff(fact_2875_inf__absorb1,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya) = Xa ) ) ).

% inf_absorb1
tff(fact_2876_inf__absorb1,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) = Xa ) ) ).

% inf_absorb1
tff(fact_2877_inf__absorb1,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya) = Xa ) ) ).

% inf_absorb1
tff(fact_2878_inf_Oabsorb2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B2,A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_2879_inf_Oabsorb2,axiom,
    ! [B2: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_2880_inf_Oabsorb2,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_2881_inf_Oabsorb2,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_2882_inf_Oabsorb2,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_2883_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = A ) ) ).

% inf.absorb1
tff(fact_2884_inf_Oabsorb1,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) = A ) ) ).

% inf.absorb1
tff(fact_2885_inf_Oabsorb1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) = A ) ) ).

% inf.absorb1
tff(fact_2886_inf_Oabsorb1,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = A ) ) ).

% inf.absorb1
tff(fact_2887_inf_Oabsorb1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = A ) ) ).

% inf.absorb1
tff(fact_2888_le__iff__inf,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Xa,Ya)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = Xa ) ) ).

% le_iff_inf
tff(fact_2889_le__iff__inf,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya) = Xa ) ) ).

% le_iff_inf
tff(fact_2890_le__iff__inf,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya) = Xa ) ) ).

% le_iff_inf
tff(fact_2891_le__iff__inf,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) = Xa ) ) ).

% le_iff_inf
tff(fact_2892_le__iff__inf,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya) = Xa ) ) ).

% le_iff_inf
tff(fact_2893_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y),X2)
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y),Y)
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( ord_le3146513528884898305at_nat(X2,Y)
             => ( ord_le3146513528884898305at_nat(X2,Z2)
               => ord_le3146513528884898305at_nat(X2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y),Z2)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Xa),Ya) ) ) ) ) ).

% inf_unique
tff(fact_2894_inf__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,Xa: set_int,Ya: set_int] :
      ( ! [X2: set_int,Y: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,X2),Y)),X2)
     => ( ! [X2: set_int,Y: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,X2),Y)),Y)
       => ( ! [X2: set_int,Y: set_int,Z2: set_int] :
              ( aa_set_int_o(ord_less_eq_set_int(X2),Y)
             => ( aa_set_int_o(ord_less_eq_set_int(X2),Z2)
               => aa_set_int_o(ord_less_eq_set_int(X2),aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y),Z2)) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,Xa),Ya) ) ) ) ) ).

% inf_unique
tff(fact_2895_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,Xa: rat,Ya: rat] :
      ( ! [X2: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y)),X2)
     => ( ! [X2: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y)),Y)
       => ( ! [X2: rat,Y: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Z2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),aa_rat_rat(aa_rat_fun_rat_rat(F,Y),Z2)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(F,Xa),Ya) ) ) ) ) ).

% inf_unique
tff(fact_2896_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,Xa: nat,Ya: nat] :
      ( ! [X2: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y)),X2)
     => ( ! [X2: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y)),Y)
       => ( ! [X2: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,Y),Z2)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(F,Xa),Ya) ) ) ) ) ).

% inf_unique
tff(fact_2897_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,Xa: int,Ya: int] :
      ( ! [X2: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y)),X2)
     => ( ! [X2: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y)),Y)
       => ( ! [X2: int,Y: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Z2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,Y),Z2)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya) = aa_int_int(aa_int_fun_int_int(F,Xa),Ya) ) ) ) ) ).

% inf_unique
tff(fact_2898_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) )
     => ord_le3146513528884898305at_nat(A,B2) ) ).

% inf.orderI
tff(fact_2899_inf_OorderI,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) )
     => aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% inf.orderI
tff(fact_2900_inf_OorderI,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% inf.orderI
tff(fact_2901_inf_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% inf.orderI
tff(fact_2902_inf_OorderI,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% inf.orderI
tff(fact_2903_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B2)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) ) ) ).

% inf.orderE
tff(fact_2904_inf_OorderE,axiom,
    ! [A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B2)
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2) ) ) ).

% inf.orderE
tff(fact_2905_inf_OorderE,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) ) ) ).

% inf.orderE
tff(fact_2906_inf_OorderE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) ) ) ).

% inf.orderE
tff(fact_2907_inf_OorderE,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) ) ) ).

% inf.orderE
tff(fact_2908_le__infI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B2,Xa)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),Xa) ) ).

% le_infI2
tff(fact_2909_le__infI2,axiom,
    ! [B2: set_int,Xa: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),Xa)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),Xa) ) ).

% le_infI2
tff(fact_2910_le__infI2,axiom,
    ! [B2: rat,Xa: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),Xa) ) ).

% le_infI2
tff(fact_2911_le__infI2,axiom,
    ! [B2: nat,Xa: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),Xa) ) ).

% le_infI2
tff(fact_2912_le__infI2,axiom,
    ! [B2: int,Xa: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),Xa) ) ).

% le_infI2
tff(fact_2913_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,Xa)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),Xa) ) ).

% le_infI1
tff(fact_2914_le__infI1,axiom,
    ! [A: set_int,Xa: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),Xa)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),Xa) ) ).

% le_infI1
tff(fact_2915_le__infI1,axiom,
    ! [A: rat,Xa: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),Xa) ) ).

% le_infI1
tff(fact_2916_le__infI1,axiom,
    ! [A: nat,Xa: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),Xa) ) ).

% le_infI1
tff(fact_2917_le__infI1,axiom,
    ! [A: int,Xa: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),Xa) ) ).

% le_infI1
tff(fact_2918_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,C2)
     => ( ord_le3146513528884898305at_nat(B2,D)
       => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D)) ) ) ).

% inf_mono
tff(fact_2919_inf__mono,axiom,
    ! [A: set_int,C2: set_int,B2: set_int,D: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),C2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),D)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C2),D)) ) ) ).

% inf_mono
tff(fact_2920_inf__mono,axiom,
    ! [A: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C2),D)) ) ) ).

% inf_mono
tff(fact_2921_inf__mono,axiom,
    ! [A: nat,C2: nat,B2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C2),D)) ) ) ).

% inf_mono
tff(fact_2922_inf__mono,axiom,
    ! [A: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C2),D)) ) ) ).

% inf_mono
tff(fact_2923_le__infI,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Xa,A)
     => ( ord_le3146513528884898305at_nat(Xa,B2)
       => ord_le3146513528884898305at_nat(Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2)) ) ) ).

% le_infI
tff(fact_2924_le__infI,axiom,
    ! [Xa: set_int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),A)
     => ( aa_set_int_o(ord_less_eq_set_int(Xa),B2)
       => aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2)) ) ) ).

% le_infI
tff(fact_2925_le__infI,axiom,
    ! [Xa: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)) ) ) ).

% le_infI
tff(fact_2926_le__infI,axiom,
    ! [Xa: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)) ) ) ).

% le_infI
tff(fact_2927_le__infI,axiom,
    ! [Xa: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)) ) ) ).

% le_infI
tff(fact_2928_le__infE,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2))
     => ~ ( ord_le3146513528884898305at_nat(Xa,A)
         => ~ ord_le3146513528884898305at_nat(Xa,B2) ) ) ).

% le_infE
tff(fact_2929_le__infE,axiom,
    ! [Xa: set_int,A: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B2))
     => ~ ( aa_set_int_o(ord_less_eq_set_int(Xa),A)
         => ~ aa_set_int_o(ord_less_eq_set_int(Xa),B2) ) ) ).

% le_infE
tff(fact_2930_le__infE,axiom,
    ! [Xa: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),B2) ) ) ).

% le_infE
tff(fact_2931_le__infE,axiom,
    ! [Xa: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),B2) ) ) ).

% le_infE
tff(fact_2932_le__infE,axiom,
    ! [Xa: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),B2) ) ) ).

% le_infE
tff(fact_2933_inf__le2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya),Ya) ).

% inf_le2
tff(fact_2934_inf__le2,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),Ya) ).

% inf_le2
tff(fact_2935_inf__le2,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya)),Ya) ).

% inf_le2
tff(fact_2936_inf__le2,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Ya) ).

% inf_le2
tff(fact_2937_inf__le2,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya)),Ya) ).

% inf_le2
tff(fact_2938_inf__le1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya),Xa) ).

% inf_le1
tff(fact_2939_inf__le1,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),Xa) ).

% inf_le1
tff(fact_2940_inf__le1,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya)),Xa) ).

% inf_le1
tff(fact_2941_inf__le1,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Xa) ).

% inf_le1
tff(fact_2942_inf__le1,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya)),Xa) ).

% inf_le1
tff(fact_2943_inf__sup__ord_I1_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya),Xa) ).

% inf_sup_ord(1)
tff(fact_2944_inf__sup__ord_I1_J,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),Xa) ).

% inf_sup_ord(1)
tff(fact_2945_inf__sup__ord_I1_J,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya)),Xa) ).

% inf_sup_ord(1)
tff(fact_2946_inf__sup__ord_I1_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Xa) ).

% inf_sup_ord(1)
tff(fact_2947_inf__sup__ord_I1_J,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya)),Xa) ).

% inf_sup_ord(1)
tff(fact_2948_inf__sup__ord_I2_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya),Ya) ).

% inf_sup_ord(2)
tff(fact_2949_inf__sup__ord_I2_J,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),Ya) ).

% inf_sup_ord(2)
tff(fact_2950_inf__sup__ord_I2_J,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya)),Ya) ).

% inf_sup_ord(2)
tff(fact_2951_inf__sup__ord_I2_J,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),Ya) ).

% inf_sup_ord(2)
tff(fact_2952_inf__sup__ord_I2_J,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya)),Ya) ).

% inf_sup_ord(2)
tff(fact_2953_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(B2,C2)
     => ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),C2) ) ).

% inf.strict_coboundedI2
tff(fact_2954_inf_Ostrict__coboundedI2,axiom,
    ! [B2: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_2955_inf_Ostrict__coboundedI2,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_2956_inf_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_2957_inf_Ostrict__coboundedI2,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_2958_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A,C2)
     => ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),C2) ) ).

% inf.strict_coboundedI1
tff(fact_2959_inf_Ostrict__coboundedI1,axiom,
    ! [A: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_2960_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_2961_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_2962_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_2963_inf_Ostrict__order__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A,B2)
    <=> ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_2964_inf_Ostrict__order__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2) )
        & ( A != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_2965_inf_Ostrict__order__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) )
        & ( A != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_2966_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_2967_inf_Ostrict__order__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) )
        & ( A != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_2968_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2))
     => ~ ( ord_le7866589430770878221at_nat(A,B2)
         => ~ ord_le7866589430770878221at_nat(A,C2) ) ) ).

% inf.strict_boundedE
tff(fact_2969_inf_Ostrict__boundedE,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,B2),C2))
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_2970_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B2),C2))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_2971_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_2972_inf_Ostrict__boundedE,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_2973_inf_Oabsorb4,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(B2,A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_2974_inf_Oabsorb4,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_2975_inf_Oabsorb4,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_2976_inf_Oabsorb4,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_2977_inf_Oabsorb4,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_2978_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A,B2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = A ) ) ).

% inf.absorb3
tff(fact_2979_inf_Oabsorb3,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2) = A ) ) ).

% inf.absorb3
tff(fact_2980_inf_Oabsorb3,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2) = A ) ) ).

% inf.absorb3
tff(fact_2981_inf_Oabsorb3,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = A ) ) ).

% inf.absorb3
tff(fact_2982_inf_Oabsorb3,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2) = A ) ) ).

% inf.absorb3
tff(fact_2983_less__infI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(B2,Xa)
     => ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),Xa) ) ).

% less_infI2
tff(fact_2984_less__infI2,axiom,
    ! [B2: real,Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2)),Xa) ) ).

% less_infI2
tff(fact_2985_less__infI2,axiom,
    ! [B2: rat,Xa: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),Xa) ) ).

% less_infI2
tff(fact_2986_less__infI2,axiom,
    ! [B2: nat,Xa: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),Xa) ) ).

% less_infI2
tff(fact_2987_less__infI2,axiom,
    ! [B2: int,Xa: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),Xa) ) ).

% less_infI2
tff(fact_2988_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A,Xa)
     => ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2),Xa) ) ).

% less_infI1
tff(fact_2989_less__infI1,axiom,
    ! [A: real,Xa: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B2)),Xa) ) ).

% less_infI1
tff(fact_2990_less__infI1,axiom,
    ! [A: rat,Xa: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B2)),Xa) ) ).

% less_infI1
tff(fact_2991_less__infI1,axiom,
    ! [A: nat,Xa: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2)),Xa) ) ).

% less_infI1
tff(fact_2992_less__infI1,axiom,
    ! [A: int,Xa: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B2)),Xa) ) ).

% less_infI1
tff(fact_2993_sup__inf__distrib2,axiom,
    ! [Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)),Xa) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_2994_sup__inf__distrib2,axiom,
    ! [Ya: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_2995_sup__inf__distrib2,axiom,
    ! [Ya: nat,Z: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_2996_sup__inf__distrib2,axiom,
    ! [Ya: set_o,Z: set_o,Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Z)),Xa) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Xa)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_2997_sup__inf__distrib2,axiom,
    ! [Ya: set_int,Z: set_int,Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z)),Xa) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Xa)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_2998_sup__inf__distrib2,axiom,
    ! [Ya: filter_nat,Z: filter_nat,Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Ya),Z)),Xa) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Xa)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Z),Xa)) ).

% sup_inf_distrib2
tff(fact_2999_sup__inf__distrib1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_3000_sup__inf__distrib1,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_3001_sup__inf__distrib1,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_3002_sup__inf__distrib1,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_3003_sup__inf__distrib1,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_3004_sup__inf__distrib1,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Ya),Z)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Z)) ).

% sup_inf_distrib1
tff(fact_3005_inf__sup__distrib2,axiom,
    ! [Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z)),Xa) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_3006_inf__sup__distrib2,axiom,
    ! [Ya: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_3007_inf__sup__distrib2,axiom,
    ! [Ya: nat,Z: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_3008_inf__sup__distrib2,axiom,
    ! [Ya: set_o,Z: set_o,Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)),Xa) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Xa)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_3009_inf__sup__distrib2,axiom,
    ! [Ya: set_int,Z: set_int,Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)),Xa) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Xa)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_3010_inf__sup__distrib2,axiom,
    ! [Ya: filter_nat,Z: filter_nat,Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)),Xa) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Ya),Xa)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Z),Xa)) ).

% inf_sup_distrib2
tff(fact_3011_inf__sup__distrib1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_3012_inf__sup__distrib1,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_3013_inf__sup__distrib1,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_3014_inf__sup__distrib1,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_3015_inf__sup__distrib1,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_3016_inf__sup__distrib1,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Ya)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Z)) ).

% inf_sup_distrib1
tff(fact_3017_distrib__imp2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Z2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_3018_distrib__imp2,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Z2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_3019_distrib__imp2,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( ! [X2: nat,Y: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Z2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_3020_distrib__imp2,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] :
      ( ! [X2: set_o,Y: set_o,Z2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X2),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Y),Z2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X2),Y)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X2),Z2))
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_3021_distrib__imp2,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( ! [X2: set_int,Y: set_int,Z2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X2),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X2),Z2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_3022_distrib__imp2,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] :
      ( ! [X2: filter_nat,Y: filter_nat,Z2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z2)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X2),Y)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X2),Z2))
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Ya)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Z)) ) ) ).

% distrib_imp2
tff(fact_3023_distrib__imp1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Z2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_3024_distrib__imp1,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Z2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_3025_distrib__imp1,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( ! [X2: nat,Y: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Z2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_3026_distrib__imp1,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] :
      ( ! [X2: set_o,Y: set_o,Z2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Y),Z2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X2),Y)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X2),Z2))
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_3027_distrib__imp1,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( ! [X2: set_int,Y: set_int,Z2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X2),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X2),Z2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_3028_distrib__imp1,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] :
      ( ! [X2: filter_nat,Y: filter_nat,Z2: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Y),Z2)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X2),Y)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X2),Z2))
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Ya),Z)) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Z)) ) ) ).

% distrib_imp1
tff(fact_3029_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)),Xa) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),Xa)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3030_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Ya: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),Xa)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3031_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Ya: set_o,Z: set_o,Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Z)),Xa) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Xa)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Z),Xa)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3032_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Ya: set_int,Z: set_int,Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z)),Xa) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Xa)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Z),Xa)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3033_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z)),Xa) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),Xa)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3034_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Ya: set_nat,Z: set_nat,Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)),Xa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),Xa)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3035_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Ya: set_o,Z: set_o,Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)),Xa) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Xa)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Z),Xa)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3036_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Ya: set_int,Z: set_int,Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)),Xa) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Xa)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Z),Xa)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3037_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3038_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3039_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3040_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3041_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3042_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3043_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3044_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3045_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_3046_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(2)
tff(fact_3047_disjoint__iff__not__equal,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat )
    <=> ! [X: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X,A2)
         => ! [Xa3: product_prod_nat_nat] :
              ( member8440522571783428010at_nat(Xa3,B)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3048_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),B)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3049_disjoint__iff__not__equal,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int )
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),B)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3050_disjoint__iff__not__equal,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o )
    <=> ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A2)
         => ! [Xa3: $o] :
              ( aa_set_o_o(member_o((Xa3)),B)
             => ( (X)
              <=> ~ (Xa3) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3051_disjoint__iff__not__equal,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat )
    <=> ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A2)
         => ! [Xa3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(Xa3),B)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3052_disjoint__iff__not__equal,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( inf_inf_set_real(A2,B) = bot_bot_set_real )
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => ! [Xa3: real] :
              ( aa_set_real_o(member_real(Xa3),B)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3053_disjoint__iff__not__equal,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat )
    <=> ! [X: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
         => ! [Xa3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa3),B)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3054_Int__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_3055_Int__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_3056_Int__empty__right,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_3057_Int__empty__right,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_3058_Int__empty__right,axiom,
    ! [A2: set_set_nat] : inf_inf_set_set_nat(A2,bot_bot_set_set_nat) = bot_bot_set_set_nat ).

% Int_empty_right
tff(fact_3059_Int__empty__right,axiom,
    ! [A2: set_real] : inf_inf_set_real(A2,bot_bot_set_real) = bot_bot_set_real ).

% Int_empty_right
tff(fact_3060_Int__empty__right,axiom,
    ! [A2: set_Extended_enat] : inf_in8357106775501769908d_enat(A2,bot_bo7653980558646680370d_enat) = bot_bo7653980558646680370d_enat ).

% Int_empty_right
tff(fact_3061_Int__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_3062_Int__empty__left,axiom,
    ! [B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_3063_Int__empty__left,axiom,
    ! [B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),B) = bot_bot_set_int ).

% Int_empty_left
tff(fact_3064_Int__empty__left,axiom,
    ! [B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),B) = bot_bot_set_o ).

% Int_empty_left
tff(fact_3065_Int__empty__left,axiom,
    ! [B: set_set_nat] : inf_inf_set_set_nat(bot_bot_set_set_nat,B) = bot_bot_set_set_nat ).

% Int_empty_left
tff(fact_3066_Int__empty__left,axiom,
    ! [B: set_real] : inf_inf_set_real(bot_bot_set_real,B) = bot_bot_set_real ).

% Int_empty_left
tff(fact_3067_Int__empty__left,axiom,
    ! [B: set_Extended_enat] : inf_in8357106775501769908d_enat(bot_bo7653980558646680370d_enat,B) = bot_bo7653980558646680370d_enat ).

% Int_empty_left
tff(fact_3068_disjoint__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat )
    <=> ! [X: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X,A2)
         => ~ member8440522571783428010at_nat(X,B) ) ) ).

% disjoint_iff
tff(fact_3069_disjoint__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ~ aa_set_nat_o(member_nat(X),B) ) ) ).

% disjoint_iff
tff(fact_3070_disjoint__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int )
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => ~ aa_set_int_o(member_int(X),B) ) ) ).

% disjoint_iff
tff(fact_3071_disjoint__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o )
    <=> ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A2)
         => ~ aa_set_o_o(member_o((X)),B) ) ) ).

% disjoint_iff
tff(fact_3072_disjoint__iff,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat )
    <=> ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A2)
         => ~ aa_set_set_nat_o(member_set_nat(X),B) ) ) ).

% disjoint_iff
tff(fact_3073_disjoint__iff,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( inf_inf_set_real(A2,B) = bot_bot_set_real )
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => ~ aa_set_real_o(member_real(X),B) ) ) ).

% disjoint_iff
tff(fact_3074_disjoint__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat )
    <=> ! [X: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
         => ~ aa_set3337841544853333262enat_o(member_Extended_enat(X),B) ) ) ).

% disjoint_iff
tff(fact_3075_Int__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X2,A2)
         => ~ member8440522571783428010at_nat(X2,B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_3076_Int__emptyI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ~ aa_set_nat_o(member_nat(X2),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_3077_Int__emptyI,axiom,
    ! [A2: set_int,B: set_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A2)
         => ~ aa_set_int_o(member_int(X2),B) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_3078_Int__emptyI,axiom,
    ! [A2: set_o,B: set_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A2)
         => ~ aa_set_o_o(member_o((X2)),B) )
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_3079_Int__emptyI,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),A2)
         => ~ aa_set_set_nat_o(member_set_nat(X2),B) )
     => ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat ) ) ).

% Int_emptyI
tff(fact_3080_Int__emptyI,axiom,
    ! [A2: set_real,B: set_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A2)
         => ~ aa_set_real_o(member_real(X2),B) )
     => ( inf_inf_set_real(A2,B) = bot_bot_set_real ) ) ).

% Int_emptyI
tff(fact_3081_Int__emptyI,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ! [X2: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
         => ~ aa_set3337841544853333262enat_o(member_Extended_enat(X2),B) )
     => ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat ) ) ).

% Int_emptyI
tff(fact_3082_Int__Collect__mono,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,P: fun_Extended_enat_o,Q: fun_Extended_enat_o] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
     => ( ! [X2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),A2)
           => ( aa_Extended_enat_o(P,X2)
             => aa_Extended_enat_o(Q,X2) ) )
       => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(inf_in8357106775501769908d_enat(A2,collec4429806609662206161d_enat(P))),inf_in8357106775501769908d_enat(B,collec4429806609662206161d_enat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3083_Int__Collect__mono,axiom,
    ! [A2: set_o,B: set_o,P: fun_o_o,Q: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B)
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
           => ( aa_o_o(P,(X2))
             => aa_o_o(Q,(X2)) ) )
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),collect_o(P))),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),collect_o(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3084_Int__Collect__mono,axiom,
    ! [A2: set_real,B: set_real,P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
           => ( aa_real_o(P,X2)
             => aa_real_o(Q,X2) ) )
       => aa_set_real_o(ord_less_eq_set_real(inf_inf_set_real(A2,collect_real(P))),inf_inf_set_real(B,collect_real(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3085_Int__Collect__mono,axiom,
    ! [A2: set_list_nat,B: set_list_nat,P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A2,B)
     => ( ! [X2: list_nat] :
            ( member_list_nat(X2,A2)
           => ( aa_list_nat_o(P,X2)
             => aa_list_nat_o(Q,X2) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A2,collect_list_nat(P)),inf_inf_set_list_nat(B,collect_list_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3086_Int__Collect__mono,axiom,
    ! [A2: set_set_nat,B: set_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
     => ( ! [X2: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X2),A2)
           => ( aa_set_nat_o(P,X2)
             => aa_set_nat_o(Q,X2) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A2,collect_set_nat(P))),inf_inf_set_set_nat(B,collect_set_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3087_Int__Collect__mono,axiom,
    ! [A2: set_nat,B: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => ( aa_nat_o(P,X2)
             => aa_nat_o(Q,X2) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(P))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3088_Int__Collect__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ord_le3146513528884898305at_nat(A2,B)
     => ( ! [X2: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X2,A2)
           => ( aa_Pro6150637604445964947_nat_o(P,X2)
             => aa_Pro6150637604445964947_nat_o(Q,X2) ) )
       => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),collec3392354462482085612at_nat(P)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),collec3392354462482085612at_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3089_Int__Collect__mono,axiom,
    ! [A2: set_int,B: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A2)
           => ( aa_int_o(P,X2)
             => aa_int_o(Q,X2) ) )
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(P))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),collect_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_3090_Int__greatest,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(C,A2)
     => ( ord_le3146513528884898305at_nat(C,B)
       => ord_le3146513528884898305at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) ) ) ).

% Int_greatest
tff(fact_3091_Int__greatest,axiom,
    ! [C: set_int,A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(C),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(C),B)
       => aa_set_int_o(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) ) ) ).

% Int_greatest
tff(fact_3092_Int__absorb2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A2,B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = A2 ) ) ).

% Int_absorb2
tff(fact_3093_Int__absorb2,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = A2 ) ) ).

% Int_absorb2
tff(fact_3094_Int__absorb1,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = B ) ) ).

% Int_absorb1
tff(fact_3095_Int__absorb1,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = B ) ) ).

% Int_absorb1
tff(fact_3096_Int__lower2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B),B) ).

% Int_lower2
tff(fact_3097_Int__lower2,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),B) ).

% Int_lower2
tff(fact_3098_Int__lower1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B),A2) ).

% Int_lower1
tff(fact_3099_Int__lower1,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),A2) ).

% Int_lower1
tff(fact_3100_Int__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A2,C)
     => ( ord_le3146513528884898305at_nat(B,D2)
       => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),D2)) ) ) ).

% Int_mono
tff(fact_3101_Int__mono,axiom,
    ! [A2: set_int,C: set_int,B: set_int,D2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),C)
     => ( aa_set_int_o(ord_less_eq_set_int(B),D2)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C),D2)) ) ) ).

% Int_mono
tff(fact_3102_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(3)
tff(fact_3103_Int__insert__right,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] :
      inf_in8357106775501769908d_enat(A2,insert_Extended_enat(A,B)) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),A2),insert_Extended_enat(A,inf_in8357106775501769908d_enat(A2,B)),inf_in8357106775501769908d_enat(A2,B)) ).

% Int_insert_right
tff(fact_3104_Int__insert__right,axiom,
    ! [A2: set_real,A: real,B: set_real] :
      inf_inf_set_real(A2,insert_real(A,B)) = $ite(aa_set_real_o(member_real(A),A2),insert_real(A,inf_inf_set_real(A2,B)),inf_inf_set_real(A2,B)) ).

% Int_insert_right
tff(fact_3105_Int__insert__right,axiom,
    ! [A2: set_set_nat,A: set_nat,B: set_set_nat] :
      inf_inf_set_set_nat(A2,insert_set_nat(A,B)) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),insert_set_nat(A,inf_inf_set_set_nat(A2,B)),inf_inf_set_set_nat(A2,B)) ).

% Int_insert_right
tff(fact_3106_Int__insert__right,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat(A,B)) = $ite(aa_set_nat_o(member_nat(A),A2),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)) ).

% Int_insert_right
tff(fact_3107_Int__insert__right,axiom,
    ! [A2: set_int,A: int,B: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int(A,B)) = $ite(aa_set_int_o(member_int(A),A2),insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) ).

% Int_insert_right
tff(fact_3108_Int__insert__right,axiom,
    ! [A2: set_o,A: $o,B: set_o] :
      aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),insert_o((A),B)) = $ite(aa_set_o_o(member_o((A)),A2),insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)) ).

% Int_insert_right
tff(fact_3109_Int__insert__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B)) = $ite(member8440522571783428010at_nat(A,A2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) ).

% Int_insert_right
tff(fact_3110_Int__insert__left,axiom,
    ! [A: extended_enat,B: set_Extended_enat,C: set_Extended_enat] :
      inf_in8357106775501769908d_enat(insert_Extended_enat(A,B),C) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),C),insert_Extended_enat(A,inf_in8357106775501769908d_enat(B,C)),inf_in8357106775501769908d_enat(B,C)) ).

% Int_insert_left
tff(fact_3111_Int__insert__left,axiom,
    ! [A: real,B: set_real,C: set_real] :
      inf_inf_set_real(insert_real(A,B),C) = $ite(aa_set_real_o(member_real(A),C),insert_real(A,inf_inf_set_real(B,C)),inf_inf_set_real(B,C)) ).

% Int_insert_left
tff(fact_3112_Int__insert__left,axiom,
    ! [A: set_nat,B: set_set_nat,C: set_set_nat] :
      inf_inf_set_set_nat(insert_set_nat(A,B),C) = $ite(aa_set_set_nat_o(member_set_nat(A),C),insert_set_nat(A,inf_inf_set_set_nat(B,C)),inf_inf_set_set_nat(B,C)) ).

% Int_insert_left
tff(fact_3113_Int__insert__left,axiom,
    ! [A: nat,B: set_nat,C: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B)),C) = $ite(aa_set_nat_o(member_nat(A),C),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).

% Int_insert_left
tff(fact_3114_Int__insert__left,axiom,
    ! [A: int,B: set_int,C: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B)),C) = $ite(aa_set_int_o(member_int(A),C),insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C)) ).

% Int_insert_left
tff(fact_3115_Int__insert__left,axiom,
    ! [A: $o,B: set_o,C: set_o] :
      aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,insert_o((A),B)),C) = $ite(aa_set_o_o(member_o((A)),C),insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C)) ).

% Int_insert_left
tff(fact_3116_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B)),C) = $ite(member8440522571783428010at_nat(A,C),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ).

% Int_insert_left
tff(fact_3117_minus__nat_Odiff__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),zero_zero_nat) = M2 ).

% minus_nat.diff_0
tff(fact_3118_diffs0__imp__equal,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2) = zero_zero_nat )
       => ( M2 = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_3119_diff__less__mono2,axiom,
    ! [M2: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2)) ) ) ).

% diff_less_mono2
tff(fact_3120_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_3121_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( M2 = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_3122_le__diff__iff,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ) ).

% le_diff_iff
tff(fact_3123_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3124_diff__le__mono,axiom,
    ! [M2: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),L)) ) ).

% diff_le_mono
tff(fact_3125_diff__le__self,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),M2) ).

% diff_le_self
tff(fact_3126_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A)),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),A) ) ) ) ).

% le_diff_iff'
tff(fact_3127_diff__le__mono2,axiom,
    ! [M2: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2)) ) ).

% diff_le_mono2
tff(fact_3128_Un__Int__distrib2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C),A2)) ).

% Un_Int_distrib2
tff(fact_3129_Un__Int__distrib2,axiom,
    ! [B: set_nat,C: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),A2)) ).

% Un_Int_distrib2
tff(fact_3130_Un__Int__distrib2,axiom,
    ! [B: set_o,C: set_o,A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C)),A2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),A2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,C),A2)) ).

% Un_Int_distrib2
tff(fact_3131_Un__Int__distrib2,axiom,
    ! [B: set_int,C: set_int,A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C)),A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),A2)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),A2)) ).

% Un_Int_distrib2
tff(fact_3132_Int__Un__distrib2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),A2)) ).

% Int_Un_distrib2
tff(fact_3133_Int__Un__distrib2,axiom,
    ! [B: set_nat,C: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),A2)) ).

% Int_Un_distrib2
tff(fact_3134_Int__Un__distrib2,axiom,
    ! [B: set_o,C: set_o,A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)),A2) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),A2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,C),A2)) ).

% Int_Un_distrib2
tff(fact_3135_Int__Un__distrib2,axiom,
    ! [B: set_int,C: set_int,A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)),A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),A2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C),A2)) ).

% Int_Un_distrib2
tff(fact_3136_Un__Int__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),C)) ).

% Un_Int_distrib
tff(fact_3137_Un__Int__distrib,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C)) ).

% Un_Int_distrib
tff(fact_3138_Un__Int__distrib,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),C)) ).

% Un_Int_distrib
tff(fact_3139_Un__Int__distrib,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),C)) ).

% Un_Int_distrib
tff(fact_3140_Int__Un__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C)) ).

% Int_Un_distrib
tff(fact_3141_Int__Un__distrib,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C)) ).

% Int_Un_distrib
tff(fact_3142_Int__Un__distrib,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),C)) ).

% Int_Un_distrib
tff(fact_3143_Int__Un__distrib,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),C)) ).

% Int_Un_distrib
tff(fact_3144_Un__Int__crazy,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C),A2)) ).

% Un_Int_crazy
tff(fact_3145_Un__Int__crazy,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),A2)) ).

% Un_Int_crazy
tff(fact_3146_Un__Int__crazy,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C))),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,C),A2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C))),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,C),A2)) ).

% Un_Int_crazy
tff(fact_3147_Un__Int__crazy,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C),A2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C))),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),A2)) ).

% Un_Int_crazy
tff(fact_3148_Diff__Int__distrib2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ).

% Diff_Int_distrib2
tff(fact_3149_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).

% Diff_Int_distrib2
tff(fact_3150_Diff__Int__distrib,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),B)) ).

% Diff_Int_distrib
tff(fact_3151_Diff__Int__distrib,axiom,
    ! [C: set_nat,A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),B)) ).

% Diff_Int_distrib
tff(fact_3152_Diff__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) ).

% Diff_Diff_Int
tff(fact_3153_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) ).

% Diff_Diff_Int
tff(fact_3154_Diff__Int2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C)),B) ).

% Diff_Int2
tff(fact_3155_Diff__Int2,axiom,
    ! [A2: set_nat,C: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C)),B) ).

% Diff_Int2
tff(fact_3156_Int__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),C)) ).

% Int_Diff
tff(fact_3157_Int__Diff,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),C)) ).

% Int_Diff
tff(fact_3158_card__Diff__subset,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B),A2)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),finite_card_set_nat(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3159_card__Diff__subset,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => ( ord_le3507040750410214029t_unit(B,A2)
       => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3160_card__Diff__subset,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => ( ord_le6045566169113846134st_nat(B,A2)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),finite_card_list_nat(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3161_card__Diff__subset,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B),A2)
       => ( finite_card_complex(minus_811609699411566653omplex(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),finite_card_complex(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3162_card__Diff__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B),A2)
       => ( finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3163_card__Diff__subset,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
       => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),finite_card_nat(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3164_card__Diff__subset,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),A2)
       => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),finite_card_int(B)) ) ) ) ).

% card_Diff_subset
tff(fact_3165_diff__card__le__card__Diff,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),finite_card_set_nat(B))),finite_card_set_nat(minus_2163939370556025621et_nat(A2,B))) ) ).

% diff_card_le_card_Diff
tff(fact_3166_diff__card__le__card__Diff,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B))),finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,B))) ) ).

% diff_card_le_card_Diff
tff(fact_3167_diff__card__le__card__Diff,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),finite_card_list_nat(B))),finite_card_list_nat(minus_7954133019191499631st_nat(A2,B))) ) ).

% diff_card_le_card_Diff
tff(fact_3168_diff__card__le__card__Diff,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),finite_card_int(B))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))) ) ).

% diff_card_le_card_Diff
tff(fact_3169_diff__card__le__card__Diff,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),finite_card_complex(B))),finite_card_complex(minus_811609699411566653omplex(A2,B))) ) ).

% diff_card_le_card_Diff
tff(fact_3170_diff__card__le__card__Diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B))),finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B))) ) ).

% diff_card_le_card_Diff
tff(fact_3171_diff__card__le__card__Diff,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),finite_card_nat(B))),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))) ) ).

% diff_card_le_card_Diff
tff(fact_3172_card__Diff__singleton__if,axiom,
    ! [A2: set_complex,Xa: complex] :
      finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(Xa),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),one_one_nat),finite_card_complex(A2)) ).

% card_Diff_singleton_if
tff(fact_3173_card__Diff__singleton__if,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit))) = $ite(member_Product_unit(Xa,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A2)),one_one_nat),finite410649719033368117t_unit(A2)) ).

% card_Diff_singleton_if
tff(fact_3174_card__Diff__singleton__if,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat))) = $ite(member_list_nat(Xa,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),one_one_nat),finite_card_list_nat(A2)) ).

% card_Diff_singleton_if
tff(fact_3175_card__Diff__singleton__if,axiom,
    ! [A2: set_int,Xa: int] :
      finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(Xa),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),one_one_nat),finite_card_int(A2)) ).

% card_Diff_singleton_if
tff(fact_3176_card__Diff__singleton__if,axiom,
    ! [A2: set_o,Xa: $o] :
      finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((Xa)),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(A2)),one_one_nat),finite_card_o(A2)) ).

% card_Diff_singleton_if
tff(fact_3177_card__Diff__singleton__if,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(Xa),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),one_one_nat),finite_card_set_nat(A2)) ).

% card_Diff_singleton_if
tff(fact_3178_card__Diff__singleton__if,axiom,
    ! [A2: set_real,Xa: real] :
      finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(Xa),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A2)),one_one_nat),finite_card_real(A2)) ).

% card_Diff_singleton_if
tff(fact_3179_card__Diff__singleton__if,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A2)),one_one_nat),finite121521170596916366d_enat(A2)) ).

% card_Diff_singleton_if
tff(fact_3180_card__Diff__singleton__if,axiom,
    ! [A2: set_nat,Xa: nat] :
      finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(Xa),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),one_one_nat),finite_card_nat(A2)) ).

% card_Diff_singleton_if
tff(fact_3181_card__Diff__singleton,axiom,
    ! [Xa: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(Xa),A2)
     => ( finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3182_card__Diff__singleton,axiom,
    ! [Xa: product_unit,A2: set_Product_unit] :
      ( member_Product_unit(Xa,A2)
     => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite410649719033368117t_unit(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3183_card__Diff__singleton,axiom,
    ! [Xa: list_nat,A2: set_list_nat] :
      ( member_list_nat(Xa,A2)
     => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3184_card__Diff__singleton,axiom,
    ! [Xa: int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),A2)
     => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3185_card__Diff__singleton,axiom,
    ! [Xa: $o,A2: set_o] :
      ( aa_set_o_o(member_o((Xa)),A2)
     => ( finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3186_card__Diff__singleton,axiom,
    ! [Xa: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3187_card__Diff__singleton,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),A2)
     => ( finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3188_card__Diff__singleton,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
     => ( finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3189_card__Diff__singleton,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A2)
     => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3190_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_0_le_iff
tff(fact_3191_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_0_le_iff
tff(fact_3192_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_0_le_iff
tff(fact_3193_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_0_le_iff
tff(fact_3194_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_3195_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_3196_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_3197_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_3198_distrib__inf__le,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Z)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z))) ).

% distrib_inf_le
tff(fact_3199_distrib__inf__le,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z))) ).

% distrib_inf_le
tff(fact_3200_distrib__inf__le,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Z))),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z))) ).

% distrib_inf_le
tff(fact_3201_distrib__inf__le,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Ya)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),Z)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Ya),Z))) ).

% distrib_inf_le
tff(fact_3202_distrib__inf__le,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Z))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z))) ).

% distrib_inf_le
tff(fact_3203_distrib__inf__le,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Ya),Z))) ).

% distrib_inf_le
tff(fact_3204_distrib__inf__le,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Ya),Z))) ).

% distrib_inf_le
tff(fact_3205_distrib__inf__le,axiom,
    ! [Xa: int,Ya: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Ya)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),aa_int_int(aa_int_fun_int_int(sup_sup_int,Ya),Z))) ).

% distrib_inf_le
tff(fact_3206_distrib__sup__le,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),Z)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Ya)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Z))) ).

% distrib_sup_le
tff(fact_3207_distrib__sup__le,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Z))) ).

% distrib_sup_le
tff(fact_3208_distrib__sup__le,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] : aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),Z))),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Z))) ).

% distrib_sup_le
tff(fact_3209_distrib__sup__le,axiom,
    ! [Xa: filter_nat,Ya: filter_nat,Z: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Ya),Z)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Ya)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Xa),Z))) ).

% distrib_sup_le
tff(fact_3210_distrib__sup__le,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),Z))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Z))) ).

% distrib_sup_le
tff(fact_3211_distrib__sup__le,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Ya),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Xa),Z))) ).

% distrib_sup_le
tff(fact_3212_distrib__sup__le,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Ya),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Xa),Z))) ).

% distrib_sup_le
tff(fact_3213_distrib__sup__le,axiom,
    ! [Xa: int,Ya: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),aa_int_int(aa_int_fun_int_int(inf_inf_int,Ya),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Ya)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Xa),Z))) ).

% distrib_sup_le
tff(fact_3214_less__imp__of__nat__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_3215_less__imp__of__nat__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_3216_less__imp__of__nat__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% less_imp_of_nat_less
tff(fact_3217_less__imp__of__nat__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% less_imp_of_nat_less
tff(fact_3218_of__nat__less__imp__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_imp_less
tff(fact_3219_of__nat__less__imp__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_imp_less
tff(fact_3220_of__nat__less__imp__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_imp_less
tff(fact_3221_of__nat__less__imp__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% of_nat_less_imp_less
tff(fact_3222_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_3223_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_3224_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_3225_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_3226_infinite__arbitrarily__large,axiom,
    ! [A2: set_set_nat,Na: nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ? [B3: set_set_nat] :
          ( aa_set_set_nat_o(finite1152437895449049373et_nat,B3)
          & ( finite_card_set_nat(B3) = Na )
          & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B3),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3227_infinite__arbitrarily__large,axiom,
    ! [A2: set_Product_unit,Na: nat] :
      ( ~ finite4290736615968046902t_unit(A2)
     => ? [B3: set_Product_unit] :
          ( finite4290736615968046902t_unit(B3)
          & ( finite410649719033368117t_unit(B3) = Na )
          & ord_le3507040750410214029t_unit(B3,A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3228_infinite__arbitrarily__large,axiom,
    ! [A2: set_list_nat,Na: nat] :
      ( ~ finite8100373058378681591st_nat(A2)
     => ? [B3: set_list_nat] :
          ( finite8100373058378681591st_nat(B3)
          & ( finite_card_list_nat(B3) = Na )
          & ord_le6045566169113846134st_nat(B3,A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3229_infinite__arbitrarily__large,axiom,
    ! [A2: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ? [B3: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B3)
          & ( finite_card_nat(B3) = Na )
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B3),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3230_infinite__arbitrarily__large,axiom,
    ! [A2: set_complex,Na: nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [B3: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B3)
          & ( finite_card_complex(B3) = Na )
          & aa_set_complex_o(ord_le211207098394363844omplex(B3),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3231_infinite__arbitrarily__large,axiom,
    ! [A2: set_Extended_enat,Na: nat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ? [B3: set_Extended_enat] :
          ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B3)
          & ( finite121521170596916366d_enat(B3) = Na )
          & aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B3),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3232_infinite__arbitrarily__large,axiom,
    ! [A2: set_int,Na: nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ? [B3: set_int] :
          ( aa_set_int_o(finite_finite_int,B3)
          & ( finite_card_int(B3) = Na )
          & aa_set_int_o(ord_less_eq_set_int(B3),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_3233_card__subset__eq,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
       => ( ( finite_card_set_nat(A2) = finite_card_set_nat(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3234_card__subset__eq,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => ( ord_le3507040750410214029t_unit(A2,B)
       => ( ( finite410649719033368117t_unit(A2) = finite410649719033368117t_unit(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3235_card__subset__eq,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => ( ord_le6045566169113846134st_nat(A2,B)
       => ( ( finite_card_list_nat(A2) = finite_card_list_nat(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3236_card__subset__eq,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => ( ( finite_card_nat(A2) = finite_card_nat(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3237_card__subset__eq,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B)
       => ( ( finite_card_complex(A2) = finite_card_complex(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3238_card__subset__eq,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
       => ( ( finite121521170596916366d_enat(A2) = finite121521170596916366d_enat(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3239_card__subset__eq,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B)
       => ( ( finite_card_int(A2) = finite_card_int(B) )
         => ( A2 = B ) ) ) ) ).

% card_subset_eq
tff(fact_3240_card__le__if__inj__on__rel,axiom,
    ! [B: set_real,A2: set_Extended_enat,R2: fun_Ex8943761750289878562real_o] :
      ( aa_set_real_o(finite_finite_real,B)
     => ( ! [A3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B)
                & aa_real_o(aa_Ext8505340146412899099real_o(R2,A3),B9) ) )
       => ( ! [A1: extended_enat,A22: extended_enat,B4: real] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A1),A2)
             => ( aa_set3337841544853333262enat_o(member_Extended_enat(A22),A2)
               => ( aa_set_real_o(member_real(B4),B)
                 => ( aa_real_o(aa_Ext8505340146412899099real_o(R2,A1),B4)
                   => ( aa_real_o(aa_Ext8505340146412899099real_o(R2,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite_card_real(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3241_card__le__if__inj__on__rel,axiom,
    ! [B: set_o,A2: set_Extended_enat,R2: fun_Ex3734807115953967292un_o_o] :
      ( aa_set_o_o(finite_finite_o,B)
     => ( ! [A3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B)
                & aa_o_o(aa_Ext1181273229397116269un_o_o(R2,A3),(B9)) ) )
       => ( ! [A1: extended_enat,A22: extended_enat,B4: $o] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A1),A2)
             => ( aa_set3337841544853333262enat_o(member_Extended_enat(A22),A2)
               => ( aa_set_o_o(member_o((B4)),B)
                 => ( aa_o_o(aa_Ext1181273229397116269un_o_o(R2,A1),(B4))
                   => ( aa_o_o(aa_Ext1181273229397116269un_o_o(R2,A22),(B4))
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite_card_o(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3242_card__le__if__inj__on__rel,axiom,
    ! [B: set_real,A2: set_real,R2: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B)
                & aa_real_o(aa_real_fun_real_o(R2,A3),B9) ) )
       => ( ! [A1: real,A22: real,B4: real] :
              ( aa_set_real_o(member_real(A1),A2)
             => ( aa_set_real_o(member_real(A22),A2)
               => ( aa_set_real_o(member_real(B4),B)
                 => ( aa_real_o(aa_real_fun_real_o(R2,A1),B4)
                   => ( aa_real_o(aa_real_fun_real_o(R2,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite_card_real(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3243_card__le__if__inj__on__rel,axiom,
    ! [B: set_o,A2: set_real,R2: fun_real_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B)
                & aa_o_o(aa_real_fun_o_o(R2,A3),(B9)) ) )
       => ( ! [A1: real,A22: real,B4: $o] :
              ( aa_set_real_o(member_real(A1),A2)
             => ( aa_set_real_o(member_real(A22),A2)
               => ( aa_set_o_o(member_o((B4)),B)
                 => ( aa_o_o(aa_real_fun_o_o(R2,A1),(B4))
                   => ( aa_o_o(aa_real_fun_o_o(R2,A22),(B4))
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite_card_o(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3244_card__le__if__inj__on__rel,axiom,
    ! [B: set_real,A2: set_int,R2: fun_int_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B)
                & aa_real_o(aa_int_fun_real_o(R2,A3),B9) ) )
       => ( ! [A1: int,A22: int,B4: real] :
              ( aa_set_int_o(member_int(A1),A2)
             => ( aa_set_int_o(member_int(A22),A2)
               => ( aa_set_real_o(member_real(B4),B)
                 => ( aa_real_o(aa_int_fun_real_o(R2,A1),B4)
                   => ( aa_real_o(aa_int_fun_real_o(R2,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_real(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3245_card__le__if__inj__on__rel,axiom,
    ! [B: set_o,A2: set_int,R2: fun_int_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B)
                & aa_o_o(aa_int_fun_o_o(R2,A3),(B9)) ) )
       => ( ! [A1: int,A22: int,B4: $o] :
              ( aa_set_int_o(member_int(A1),A2)
             => ( aa_set_int_o(member_int(A22),A2)
               => ( aa_set_o_o(member_o((B4)),B)
                 => ( aa_o_o(aa_int_fun_o_o(R2,A1),(B4))
                   => ( aa_o_o(aa_int_fun_o_o(R2,A22),(B4))
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_o(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3246_card__le__if__inj__on__rel,axiom,
    ! [B: set_real,A2: set_o,R2: fun_o_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A2)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B)
                & aa_real_o(aa_o_fun_real_o(R2,(A3)),B9) ) )
       => ( ! [A1: $o,A22: $o,B4: real] :
              ( aa_set_o_o(member_o((A1)),A2)
             => ( aa_set_o_o(member_o((A22)),A2)
               => ( aa_set_real_o(member_real(B4),B)
                 => ( aa_real_o(aa_o_fun_real_o(R2,(A1)),B4)
                   => ( aa_real_o(aa_o_fun_real_o(R2,(A22)),B4)
                     => ( (A1)
                      <=> (A22) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A2)),finite_card_real(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3247_card__le__if__inj__on__rel,axiom,
    ! [B: set_o,A2: set_o,R2: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A2)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B)
                & aa_o_o(aa_o_fun_o_o(R2,(A3)),(B9)) ) )
       => ( ! [A1: $o,A22: $o,B4: $o] :
              ( aa_set_o_o(member_o((A1)),A2)
             => ( aa_set_o_o(member_o((A22)),A2)
               => ( aa_set_o_o(member_o((B4)),B)
                 => ( aa_o_o(aa_o_fun_o_o(R2,(A1)),(B4))
                   => ( aa_o_o(aa_o_fun_o_o(R2,(A22)),(B4))
                     => ( (A1)
                      <=> (A22) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A2)),finite_card_o(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3248_card__le__if__inj__on__rel,axiom,
    ! [B: set_Product_unit,A2: set_Extended_enat,R2: fun_Ex5682132948637686023unit_o] :
      ( finite4290736615968046902t_unit(B)
     => ( ! [A3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
           => ? [B9: product_unit] :
                ( member_Product_unit(B9,B)
                & aa_Product_unit_o(aa_Ext9056830302645520696unit_o(R2,A3),B9) ) )
       => ( ! [A1: extended_enat,A22: extended_enat,B4: product_unit] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A1),A2)
             => ( aa_set3337841544853333262enat_o(member_Extended_enat(A22),A2)
               => ( member_Product_unit(B4,B)
                 => ( aa_Product_unit_o(aa_Ext9056830302645520696unit_o(R2,A1),B4)
                   => ( aa_Product_unit_o(aa_Ext9056830302645520696unit_o(R2,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite410649719033368117t_unit(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3249_card__le__if__inj__on__rel,axiom,
    ! [B: set_Product_unit,A2: set_real,R2: fun_re6928409447419994551unit_o] :
      ( finite4290736615968046902t_unit(B)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ? [B9: product_unit] :
                ( member_Product_unit(B9,B)
                & aa_Product_unit_o(aa_rea3180783709397473784unit_o(R2,A3),B9) ) )
       => ( ! [A1: real,A22: real,B4: product_unit] :
              ( aa_set_real_o(member_real(A1),A2)
             => ( aa_set_real_o(member_real(A22),A2)
               => ( member_Product_unit(B4,B)
                 => ( aa_Product_unit_o(aa_rea3180783709397473784unit_o(R2,A1),B4)
                   => ( aa_Product_unit_o(aa_rea3180783709397473784unit_o(R2,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite410649719033368117t_unit(B)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3250_of__nat__diff,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ) ).

% of_nat_diff
tff(fact_3251_of__nat__diff,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ) ).

% of_nat_diff
tff(fact_3252_of__nat__diff,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ) ).

% of_nat_diff
tff(fact_3253_of__nat__diff,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% of_nat_diff
tff(fact_3254_card__insert__le,axiom,
    ! [A2: set_int,Xa: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_int(insert_int(Xa,A2))) ).

% card_insert_le
tff(fact_3255_card__insert__le,axiom,
    ! [A2: set_o,Xa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A2)),finite_card_o(insert_o((Xa),A2))) ).

% card_insert_le
tff(fact_3256_card__insert__le,axiom,
    ! [A2: set_real,Xa: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite_card_real(insert_real(Xa,A2))) ).

% card_insert_le
tff(fact_3257_card__insert__le,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(insert_Extended_enat(Xa,A2))) ).

% card_insert_le
tff(fact_3258_card__insert__le,axiom,
    ! [A2: set_nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(A2)),finite_card_nat(insert_nat(Xa,A2))) ).

% card_insert_le
tff(fact_3259_card__insert__le,axiom,
    ! [A2: set_complex,Xa: complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_complex(insert_complex(Xa,A2))) ).

% card_insert_le
tff(fact_3260_card__insert__le,axiom,
    ! [A2: set_set_nat,Xa: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),finite_card_set_nat(insert_set_nat(Xa,A2))) ).

% card_insert_le
tff(fact_3261_card__insert__le,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(insert_Product_unit(Xa,A2))) ).

% card_insert_le
tff(fact_3262_card__insert__le,axiom,
    ! [A2: set_list_nat,Xa: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),finite_card_list_nat(insert_list_nat(Xa,A2))) ).

% card_insert_le
tff(fact_3263_Diff__triv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B) = A2 ) ) ).

% Diff_triv
tff(fact_3264_Diff__triv,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B) = A2 ) ) ).

% Diff_triv
tff(fact_3265_Diff__triv,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o )
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B) = A2 ) ) ).

% Diff_triv
tff(fact_3266_Diff__triv,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat )
     => ( minus_2163939370556025621et_nat(A2,B) = A2 ) ) ).

% Diff_triv
tff(fact_3267_Diff__triv,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( inf_inf_set_real(A2,B) = bot_bot_set_real )
     => ( minus_minus_set_real(A2,B) = A2 ) ) ).

% Diff_triv
tff(fact_3268_Diff__triv,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat )
     => ( minus_925952699566721837d_enat(A2,B) = A2 ) ) ).

% Diff_triv
tff(fact_3269_Diff__triv,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B) = A2 ) ) ).

% Diff_triv
tff(fact_3270_Int__Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_3271_Int__Diff__disjoint,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_3272_Int__Diff__disjoint,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_3273_Int__Diff__disjoint,axiom,
    ! [A2: set_set_nat,B: set_set_nat] : inf_inf_set_set_nat(inf_inf_set_set_nat(A2,B),minus_2163939370556025621et_nat(A2,B)) = bot_bot_set_set_nat ).

% Int_Diff_disjoint
tff(fact_3274_Int__Diff__disjoint,axiom,
    ! [A2: set_real,B: set_real] : inf_inf_set_real(inf_inf_set_real(A2,B),minus_minus_set_real(A2,B)) = bot_bot_set_real ).

% Int_Diff_disjoint
tff(fact_3275_Int__Diff__disjoint,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] : inf_in8357106775501769908d_enat(inf_in8357106775501769908d_enat(A2,B),minus_925952699566721837d_enat(A2,B)) = bot_bo7653980558646680370d_enat ).

% Int_Diff_disjoint
tff(fact_3276_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_3277_Un__Int__assoc__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C)) )
    <=> ord_le3146513528884898305at_nat(C,A2) ) ).

% Un_Int_assoc_eq
tff(fact_3278_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A2) ) ).

% Un_Int_assoc_eq
tff(fact_3279_Un__Int__assoc__eq,axiom,
    ! [A2: set_o,B: set_o,C: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)),C) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) )
    <=> aa_set_o_o(ord_less_eq_set_o(C),A2) ) ).

% Un_Int_assoc_eq
tff(fact_3280_Un__Int__assoc__eq,axiom,
    ! [A2: set_int,B: set_int,C: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),C) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) )
    <=> aa_set_int_o(ord_less_eq_set_int(C),A2) ) ).

% Un_Int_assoc_eq
tff(fact_3281_diff__less,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),M2) ) ) ).

% diff_less
tff(fact_3282_less__diff__iff,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ) ).

% less_diff_iff
tff(fact_3283_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2)) ) ) ).

% diff_less_mono
tff(fact_3284_Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C)) ).

% Diff_Un
tff(fact_3285_Diff__Un,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B),C)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),C)) ).

% Diff_Un
tff(fact_3286_Diff__Un,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),C)) ).

% Diff_Un
tff(fact_3287_Diff__Un,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C)) ).

% Diff_Un
tff(fact_3288_Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C)) ).

% Diff_Int
tff(fact_3289_Diff__Int,axiom,
    ! [A2: set_o,B: set_o,C: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,B),C)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),C)) ).

% Diff_Int
tff(fact_3290_Diff__Int,axiom,
    ! [A2: set_int,B: set_int,C: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),C)) ).

% Diff_Int
tff(fact_3291_Diff__Int,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C)) ).

% Diff_Int
tff(fact_3292_Int__Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)) = A2 ).

% Int_Diff_Un
tff(fact_3293_Int__Diff__Un,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)) = A2 ).

% Int_Diff_Un
tff(fact_3294_Int__Diff__Un,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) = A2 ).

% Int_Diff_Un
tff(fact_3295_Int__Diff__Un,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = A2 ).

% Int_Diff_Un
tff(fact_3296_Un__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) = A2 ).

% Un_Diff_Int
tff(fact_3297_Un__Diff__Int,axiom,
    ! [A2: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)) = A2 ).

% Un_Diff_Int
tff(fact_3298_Un__Diff__Int,axiom,
    ! [A2: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) = A2 ).

% Un_Diff_Int
tff(fact_3299_Un__Diff__Int,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)) = A2 ).

% Un_Diff_Int
tff(fact_3300_Inf__fin_Oin__idem,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,Xa),lattic2677971596711400399n_real(A2)) = lattic2677971596711400399n_real(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3301_Inf__fin_Oin__idem,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),lattic3014633134055518761et_nat(A2)) = lattic3014633134055518761et_nat(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3302_Inf__fin_Oin__idem,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( aa_o_o(aa_o_fun_o_o(inf_inf_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
        <=> aa_set_o_o(lattic4107685809792843317_fin_o,A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3303_Inf__fin_Oin__idem,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),lattic5235898064620869839in_int(A2)) = lattic5235898064620869839in_int(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3304_Inf__fin_Oin__idem,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( inf_in1870772243966228564d_enat(Xa,lattic974744108425517955d_enat(A2)) = lattic974744108425517955d_enat(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3305_Inf__fin_Oin__idem,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),lattic5238388535129920115in_nat(A2)) = lattic5238388535129920115in_nat(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3306_Inf__fin_Oin__idem,axiom,
    ! [A2: set_se7855581050983116737at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( member2643936169264416010at_nat(Xa,A2)
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),lattic30941717366863870at_nat(A2)) = lattic30941717366863870at_nat(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3307_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(insert_int(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3308_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(insert_o((Xa),Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3309_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(insert_real(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3310_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_Extended_enat,Xa: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(insert_Extended_enat(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3311_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(insert_nat(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3312_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_complex,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(insert_complex(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3313_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(insert_set_nat(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3314_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_Product_unit,Xa: product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(insert_Product_unit(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3315_card__insert__le__m1,axiom,
    ! [Na: nat,Ya: set_list_nat,Xa: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(Ya)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(insert_list_nat(Xa,Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3316_is__singleton__altdef,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ( finite_card_nat(A2) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3317_is__singleton__altdef,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(is_singleton_complex,A2)
    <=> ( finite_card_complex(A2) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3318_is__singleton__altdef,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(is_singleton_set_nat,A2)
    <=> ( finite_card_set_nat(A2) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3319_is__singleton__altdef,axiom,
    ! [A2: set_Product_unit] :
      ( is_sin2160648248035936513t_unit(A2)
    <=> ( finite410649719033368117t_unit(A2) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3320_is__singleton__altdef,axiom,
    ! [A2: set_list_nat] :
      ( is_sin2641923865335537900st_nat(A2)
    <=> ( finite_card_list_nat(A2) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3321_card__eq__0__iff,axiom,
    ! [A2: set_Product_unit] :
      ( ( finite410649719033368117t_unit(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bo3957492148770167129t_unit )
        | ~ finite4290736615968046902t_unit(A2) ) ) ).

% card_eq_0_iff
tff(fact_3322_card__eq__0__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_list_nat )
        | ~ finite8100373058378681591st_nat(A2) ) ) ).

% card_eq_0_iff
tff(fact_3323_card__eq__0__iff,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_complex )
        | ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% card_eq_0_iff
tff(fact_3324_card__eq__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_card_nat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        | ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% card_eq_0_iff
tff(fact_3325_card__eq__0__iff,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_int )
        | ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% card_eq_0_iff
tff(fact_3326_card__eq__0__iff,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_o )
        | ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% card_eq_0_iff
tff(fact_3327_card__eq__0__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_set_nat )
        | ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ) ).

% card_eq_0_iff
tff(fact_3328_card__eq__0__iff,axiom,
    ! [A2: set_real] :
      ( ( finite_card_real(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_real )
        | ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% card_eq_0_iff
tff(fact_3329_card__eq__0__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bo7653980558646680370d_enat )
        | ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% card_eq_0_iff
tff(fact_3330_card__ge__0__finite,axiom,
    ! [A2: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A2))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ).

% card_ge_0_finite
tff(fact_3331_card__ge__0__finite,axiom,
    ! [A2: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A2))
     => finite4290736615968046902t_unit(A2) ) ).

% card_ge_0_finite
tff(fact_3332_card__ge__0__finite,axiom,
    ! [A2: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A2))
     => finite8100373058378681591st_nat(A2) ) ).

% card_ge_0_finite
tff(fact_3333_card__ge__0__finite,axiom,
    ! [A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_nat(A2))
     => aa_set_nat_o(finite_finite_nat,A2) ) ).

% card_ge_0_finite
tff(fact_3334_card__ge__0__finite,axiom,
    ! [A2: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A2))
     => aa_set_int_o(finite_finite_int,A2) ) ).

% card_ge_0_finite
tff(fact_3335_card__ge__0__finite,axiom,
    ! [A2: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A2))
     => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ).

% card_ge_0_finite
tff(fact_3336_card__ge__0__finite,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite121521170596916366d_enat(A2))
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ).

% card_ge_0_finite
tff(fact_3337_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_set_nat,C: nat] :
      ( ! [G2: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(G2),F2)
         => ( aa_set_set_nat_o(finite1152437895449049373et_nat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(G2)),C) ) )
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3338_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Product_unit,C: nat] :
      ( ! [G2: set_Product_unit] :
          ( ord_le3507040750410214029t_unit(G2,F2)
         => ( finite4290736615968046902t_unit(G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(G2)),C) ) )
     => ( finite4290736615968046902t_unit(F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3339_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_list_nat,C: nat] :
      ( ! [G2: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G2,F2)
         => ( finite8100373058378681591st_nat(G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(G2)),C) ) )
     => ( finite8100373058378681591st_nat(F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3340_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C: nat] :
      ( ! [G2: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,G2),F2)
         => ( aa_set_nat_o(finite_finite_nat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(G2)),C) ) )
     => ( aa_set_nat_o(finite_finite_nat,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3341_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_complex,C: nat] :
      ( ! [G2: set_complex] :
          ( aa_set_complex_o(ord_le211207098394363844omplex(G2),F2)
         => ( aa_set_complex_o(finite3207457112153483333omplex,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(G2)),C) ) )
     => ( aa_set_complex_o(finite3207457112153483333omplex,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3342_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Extended_enat,C: nat] :
      ( ! [G2: set_Extended_enat] :
          ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(G2),F2)
         => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(G2)),C) ) )
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3343_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_int,C: nat] :
      ( ! [G2: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(G2),F2)
         => ( aa_set_int_o(finite_finite_int,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(G2)),C) ) )
     => ( aa_set_int_o(finite_finite_int,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(F2)),C) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3344_card__seteq,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(B)),finite_card_set_nat(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3345_card__seteq,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => ( ord_le3507040750410214029t_unit(A2,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(B)),finite410649719033368117t_unit(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3346_card__seteq,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => ( ord_le6045566169113846134st_nat(A2,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(B)),finite_card_list_nat(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3347_card__seteq,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(B)),finite_card_nat(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3348_card__seteq,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B)),finite_card_complex(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3349_card__seteq,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(B)),finite121521170596916366d_enat(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3350_card__seteq,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B)),finite_card_int(A2))
         => ( A2 = B ) ) ) ) ).

% card_seteq
tff(fact_3351_card__mono,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),finite_card_set_nat(B)) ) ) ).

% card_mono
tff(fact_3352_card__mono,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => ( ord_le3507040750410214029t_unit(A2,B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B)) ) ) ).

% card_mono
tff(fact_3353_card__mono,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => ( ord_le6045566169113846134st_nat(A2,B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),finite_card_list_nat(B)) ) ) ).

% card_mono
tff(fact_3354_card__mono,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(A2)),finite_card_nat(B)) ) ) ).

% card_mono
tff(fact_3355_card__mono,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_complex(B)) ) ) ).

% card_mono
tff(fact_3356_card__mono,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B)) ) ) ).

% card_mono
tff(fact_3357_card__mono,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_int(B)) ) ) ).

% card_mono
tff(fact_3358_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_set_nat(S))
     => ~ ! [T3: set_set_nat] :
            ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(T3),S)
           => ( ( finite_card_set_nat(T3) = Na )
             => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3359_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite410649719033368117t_unit(S))
     => ~ ! [T3: set_Product_unit] :
            ( ord_le3507040750410214029t_unit(T3,S)
           => ( ( finite410649719033368117t_unit(T3) = Na )
             => ~ finite4290736615968046902t_unit(T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3360_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_list_nat(S))
     => ~ ! [T3: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T3,S)
           => ( ( finite_card_list_nat(T3) = Na )
             => ~ finite8100373058378681591st_nat(T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3361_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_nat(S))
     => ~ ! [T3: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,T3),S)
           => ( ( finite_card_nat(T3) = Na )
             => ~ aa_set_nat_o(finite_finite_nat,T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3362_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_complex(S))
     => ~ ! [T3: set_complex] :
            ( aa_set_complex_o(ord_le211207098394363844omplex(T3),S)
           => ( ( finite_card_complex(T3) = Na )
             => ~ aa_set_complex_o(finite3207457112153483333omplex,T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3363_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite121521170596916366d_enat(S))
     => ~ ! [T3: set_Extended_enat] :
            ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(T3),S)
           => ( ( finite121521170596916366d_enat(T3) = Na )
             => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3364_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_int(S))
     => ~ ! [T3: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(T3),S)
           => ( ( finite_card_int(T3) = Na )
             => ~ aa_set_int_o(finite_finite_int,T3) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3365_card__less__sym__Diff,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A2)),finite_card_set_nat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B))),finite_card_set_nat(minus_2163939370556025621et_nat(B,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3366_card__less__sym__Diff,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( finite4290736615968046902t_unit(B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,B))),finite410649719033368117t_unit(minus_6452836326544984404t_unit(B,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3367_card__less__sym__Diff,axiom,
    ! [A2: set_list_nat,B: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A2)),finite_card_list_nat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B))),finite_card_list_nat(minus_7954133019191499631st_nat(B,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3368_card__less__sym__Diff,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A2)),finite_card_int(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3369_card__less__sym__Diff,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A2)),finite_card_complex(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A2,B))),finite_card_complex(minus_811609699411566653omplex(B,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3370_card__less__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B))),finite121521170596916366d_enat(minus_925952699566721837d_enat(B,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3371_card__less__sym__Diff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(A2)),finite_card_nat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3372_card__le__sym__Diff,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),finite_card_set_nat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B))),finite_card_set_nat(minus_2163939370556025621et_nat(B,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3373_card__le__sym__Diff,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( finite4290736615968046902t_unit(B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,B))),finite410649719033368117t_unit(minus_6452836326544984404t_unit(B,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3374_card__le__sym__Diff,axiom,
    ! [A2: set_list_nat,B: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),finite_card_list_nat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B))),finite_card_list_nat(minus_7954133019191499631st_nat(B,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3375_card__le__sym__Diff,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_int(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B),A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3376_card__le__sym__Diff,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_complex(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A2,B))),finite_card_complex(minus_811609699411566653omplex(B,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3377_card__le__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B))),finite121521170596916366d_enat(minus_925952699566721837d_enat(B,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3378_card__le__sym__Diff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(A2)),finite_card_nat(B))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B))),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3379_card__1__singletonE,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = one_one_nat )
     => ~ ! [X2: complex] : A2 != insert_complex(X2,bot_bot_set_complex) ) ).

% card_1_singletonE
tff(fact_3380_card__1__singletonE,axiom,
    ! [A2: set_Product_unit] :
      ( ( finite410649719033368117t_unit(A2) = one_one_nat )
     => ~ ! [X2: product_unit] : A2 != insert_Product_unit(X2,bot_bo3957492148770167129t_unit) ) ).

% card_1_singletonE
tff(fact_3381_card__1__singletonE,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = one_one_nat )
     => ~ ! [X2: list_nat] : A2 != insert_list_nat(X2,bot_bot_set_list_nat) ) ).

% card_1_singletonE
tff(fact_3382_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( finite_card_nat(A2) = one_one_nat )
     => ~ ! [X2: nat] : A2 != insert_nat(X2,bot_bot_set_nat) ) ).

% card_1_singletonE
tff(fact_3383_card__1__singletonE,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = one_one_nat )
     => ~ ! [X2: int] : A2 != insert_int(X2,bot_bot_set_int) ) ).

% card_1_singletonE
tff(fact_3384_card__1__singletonE,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = one_one_nat )
     => ~ ! [X2: $o] : A2 != insert_o((X2),bot_bot_set_o) ) ).

% card_1_singletonE
tff(fact_3385_card__1__singletonE,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = one_one_nat )
     => ~ ! [X2: set_nat] : A2 != insert_set_nat(X2,bot_bot_set_set_nat) ) ).

% card_1_singletonE
tff(fact_3386_card__1__singletonE,axiom,
    ! [A2: set_real] :
      ( ( finite_card_real(A2) = one_one_nat )
     => ~ ! [X2: real] : A2 != insert_real(X2,bot_bot_set_real) ) ).

% card_1_singletonE
tff(fact_3387_card__1__singletonE,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(A2) = one_one_nat )
     => ~ ! [X2: extended_enat] : A2 != insert_Extended_enat(X2,bot_bo7653980558646680370d_enat) ) ).

% card_1_singletonE
tff(fact_3388_psubset__card__mono,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A2)),finite_card_set_nat(B)) ) ) ).

% psubset_card_mono
tff(fact_3389_psubset__card__mono,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => ( ord_le8056459307392131481t_unit(A2,B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B)) ) ) ).

% psubset_card_mono
tff(fact_3390_psubset__card__mono,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => ( ord_le1190675801316882794st_nat(A2,B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A2)),finite_card_list_nat(B)) ) ) ).

% psubset_card_mono
tff(fact_3391_psubset__card__mono,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(A2)),finite_card_nat(B)) ) ) ).

% psubset_card_mono
tff(fact_3392_psubset__card__mono,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_set_int(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A2)),finite_card_int(B)) ) ) ).

% psubset_card_mono
tff(fact_3393_psubset__card__mono,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_less_set_complex(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A2)),finite_card_complex(B)) ) ) ).

% psubset_card_mono
tff(fact_3394_psubset__card__mono,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B)) ) ) ).

% psubset_card_mono
tff(fact_3395_card__gt__0__iff,axiom,
    ! [A2: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A2))
    <=> ( ( A2 != bot_bo3957492148770167129t_unit )
        & finite4290736615968046902t_unit(A2) ) ) ).

% card_gt_0_iff
tff(fact_3396_card__gt__0__iff,axiom,
    ! [A2: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A2))
    <=> ( ( A2 != bot_bot_set_list_nat )
        & finite8100373058378681591st_nat(A2) ) ) ).

% card_gt_0_iff
tff(fact_3397_card__gt__0__iff,axiom,
    ! [A2: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A2))
    <=> ( ( A2 != bot_bot_set_complex )
        & aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% card_gt_0_iff
tff(fact_3398_card__gt__0__iff,axiom,
    ! [A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_nat(A2))
    <=> ( ( A2 != bot_bot_set_nat )
        & aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% card_gt_0_iff
tff(fact_3399_card__gt__0__iff,axiom,
    ! [A2: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A2))
    <=> ( ( A2 != bot_bot_set_int )
        & aa_set_int_o(finite_finite_int,A2) ) ) ).

% card_gt_0_iff
tff(fact_3400_card__gt__0__iff,axiom,
    ! [A2: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A2))
    <=> ( ( A2 != bot_bot_set_o )
        & aa_set_o_o(finite_finite_o,A2) ) ) ).

% card_gt_0_iff
tff(fact_3401_card__gt__0__iff,axiom,
    ! [A2: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A2))
    <=> ( ( A2 != bot_bot_set_set_nat )
        & aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ) ).

% card_gt_0_iff
tff(fact_3402_card__gt__0__iff,axiom,
    ! [A2: set_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_real(A2))
    <=> ( ( A2 != bot_bot_set_real )
        & aa_set_real_o(finite_finite_real,A2) ) ) ).

% card_gt_0_iff
tff(fact_3403_card__gt__0__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite121521170596916366d_enat(A2))
    <=> ( ( A2 != bot_bo7653980558646680370d_enat )
        & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% card_gt_0_iff
tff(fact_3404_card__Diff1__le,axiom,
    ! [A2: set_complex,Xa: complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))),finite_card_complex(A2)) ).

% card_Diff1_le
tff(fact_3405_card__Diff1__le,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A2)) ).

% card_Diff1_le
tff(fact_3406_card__Diff1__le,axiom,
    ! [A2: set_list_nat,Xa: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)))),finite_card_list_nat(A2)) ).

% card_Diff1_le
tff(fact_3407_card__Diff1__le,axiom,
    ! [A2: set_int,Xa: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)))),finite_card_int(A2)) ).

% card_Diff1_le
tff(fact_3408_card__Diff1__le,axiom,
    ! [A2: set_o,Xa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)))),finite_card_o(A2)) ).

% card_Diff1_le
tff(fact_3409_card__Diff1__le,axiom,
    ! [A2: set_set_nat,Xa: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)))),finite_card_set_nat(A2)) ).

% card_Diff1_le
tff(fact_3410_card__Diff1__le,axiom,
    ! [A2: set_real,Xa: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))),finite_card_real(A2)) ).

% card_Diff1_le
tff(fact_3411_card__Diff1__le,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A2)) ).

% card_Diff1_le
tff(fact_3412_card__Diff1__le,axiom,
    ! [A2: set_nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)))),finite_card_nat(A2)) ).

% card_Diff1_le
tff(fact_3413_card__psubset,axiom,
    ! [B: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A2)),finite_card_set_nat(B))
         => aa_set_set_nat_o(ord_less_set_set_nat(A2),B) ) ) ) ).

% card_psubset
tff(fact_3414_card__psubset,axiom,
    ! [B: set_Product_unit,A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(B)
     => ( ord_le3507040750410214029t_unit(A2,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B))
         => ord_le8056459307392131481t_unit(A2,B) ) ) ) ).

% card_psubset
tff(fact_3415_card__psubset,axiom,
    ! [B: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B)
     => ( ord_le6045566169113846134st_nat(A2,B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A2)),finite_card_list_nat(B))
         => ord_le1190675801316882794st_nat(A2,B) ) ) ) ).

% card_psubset
tff(fact_3416_card__psubset,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(A2)),finite_card_nat(B))
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B) ) ) ) ).

% card_psubset
tff(fact_3417_card__psubset,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A2)),finite_card_complex(B))
         => aa_set_complex_o(ord_less_set_complex(A2),B) ) ) ) ).

% card_psubset
tff(fact_3418_card__psubset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B))
         => aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),B) ) ) ) ).

% card_psubset
tff(fact_3419_card__psubset,axiom,
    ! [B: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A2)),finite_card_int(B))
         => aa_set_int_o(ord_less_set_int(A2),B) ) ) ) ).

% card_psubset
tff(fact_3420_Inf__fin_Osubset,axiom,
    ! [A2: set_se7855581050983116737at_nat,B: set_se7855581050983116737at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( B != bot_bo3083307316010499117at_nat )
       => ( ord_le2077887516847798113at_nat(B,A2)
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,lattic30941717366863870at_nat(B)),lattic30941717366863870at_nat(A2)) = lattic30941717366863870at_nat(A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3421_Inf__fin_Osubset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( B != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,lattic5238388535129920115in_nat(B)),lattic5238388535129920115in_nat(A2)) = lattic5238388535129920115in_nat(A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3422_Inf__fin_Osubset,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( B != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B),A2)
         => ( aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,B)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
          <=> aa_set_o_o(lattic4107685809792843317_fin_o,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3423_Inf__fin_Osubset,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( B != bot_bot_set_set_nat )
       => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B),A2)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,lattic3014633134055518761et_nat(B)),lattic3014633134055518761et_nat(A2)) = lattic3014633134055518761et_nat(A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3424_Inf__fin_Osubset,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( B != bot_bot_set_real )
       => ( aa_set_real_o(ord_less_eq_set_real(B),A2)
         => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,lattic2677971596711400399n_real(B)),lattic2677971596711400399n_real(A2)) = lattic2677971596711400399n_real(A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3425_Inf__fin_Osubset,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( B != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B),A2)
         => ( inf_in1870772243966228564d_enat(lattic974744108425517955d_enat(B),lattic974744108425517955d_enat(A2)) = lattic974744108425517955d_enat(A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3426_Inf__fin_Osubset,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( B != bot_bot_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(B),A2)
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,lattic5235898064620869839in_int(B)),lattic5235898064620869839in_int(A2)) = lattic5235898064620869839in_int(A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3427_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_se7855581050983116737at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ~ member2643936169264416010at_nat(Xa,A2)
       => ( ( A2 != bot_bo3083307316010499117at_nat )
         => ( lattic30941717366863870at_nat(insert9200635055090092081at_nat(Xa,A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),lattic30941717366863870at_nat(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3428_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( lattic5238388535129920115in_nat(insert_nat(Xa,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Xa),lattic5238388535129920115in_nat(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3429_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xa),A2)
       => ( ( A2 != bot_bot_set_int )
         => ( lattic5235898064620869839in_int(insert_int(Xa,A2)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,Xa),lattic5235898064620869839in_int(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3430_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((Xa)),A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((Xa),A2))
          <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(Xa)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3431_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( ( A2 != bot_bot_set_set_nat )
         => ( lattic3014633134055518761et_nat(insert_set_nat(Xa,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),lattic3014633134055518761et_nat(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3432_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xa),A2)
       => ( ( A2 != bot_bot_set_real )
         => ( lattic2677971596711400399n_real(insert_real(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,Xa),lattic2677971596711400399n_real(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3433_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( lattic974744108425517955d_enat(insert_Extended_enat(Xa,A2)) = inf_in1870772243966228564d_enat(Xa,lattic974744108425517955d_enat(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3434_Inf__fin_Oclosed,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( A2 != bot_bo3083307316010499117at_nat )
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : member2643936169264416010at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y),insert9200635055090092081at_nat(X2,insert9200635055090092081at_nat(Y,bot_bo3083307316010499117at_nat)))
         => member2643936169264416010at_nat(lattic30941717366863870at_nat(A2),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3435_Inf__fin_Oclosed,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat,Y: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y)),insert_nat(X2,insert_nat(Y,bot_bot_set_nat)))
         => aa_set_nat_o(member_nat(lattic5238388535129920115in_nat(A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3436_Inf__fin_Oclosed,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int,Y: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(inf_inf_int,X2),Y)),insert_int(X2,insert_int(Y,bot_bot_set_int)))
         => aa_set_int_o(member_int(lattic5235898064620869839in_int(A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3437_Inf__fin_Oclosed,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o,Y: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(inf_inf_o,(X2)),(Y))),insert_o((X2),insert_o((Y),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic4107685809792843317_fin_o,A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3438_Inf__fin_Oclosed,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y: set_nat] : aa_set_set_nat_o(member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y)),insert_set_nat(X2,insert_set_nat(Y,bot_bot_set_set_nat)))
         => aa_set_set_nat_o(member_set_nat(lattic3014633134055518761et_nat(A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3439_Inf__fin_Oclosed,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real,Y: real] : aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,X2),Y)),insert_real(X2,insert_real(Y,bot_bot_set_real)))
         => aa_set_real_o(member_real(lattic2677971596711400399n_real(A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3440_Inf__fin_Oclosed,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X2: extended_enat,Y: extended_enat] : aa_set3337841544853333262enat_o(member_Extended_enat(inf_in1870772243966228564d_enat(X2,Y)),insert_Extended_enat(X2,insert_Extended_enat(Y,bot_bo7653980558646680370d_enat)))
         => aa_set3337841544853333262enat_o(member_Extended_enat(lattic974744108425517955d_enat(A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_3441_Inf__fin_Ounion,axiom,
    ! [A2: set_se7855581050983116737at_nat,B: set_se7855581050983116737at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( A2 != bot_bo3083307316010499117at_nat )
       => ( finite9047747110432174090at_nat(B)
         => ( ( B != bot_bo3083307316010499117at_nat )
           => ( lattic30941717366863870at_nat(sup_su3642409539654194069at_nat(A2,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,lattic30941717366863870at_nat(A2)),lattic30941717366863870at_nat(B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3442_Inf__fin_Ounion,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B)
         => ( ( B != bot_bot_set_nat )
           => ( lattic5238388535129920115in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,lattic5238388535129920115in_nat(A2)),lattic5238388535129920115in_nat(B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3443_Inf__fin_Ounion,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B)
         => ( ( B != bot_bot_set_int )
           => ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,lattic5235898064620869839in_int(A2)),lattic5235898064620869839in_int(B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3444_Inf__fin_Ounion,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B)
         => ( ( B != bot_bot_set_o )
           => ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))
            <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,A2)),aa_set_o_o(lattic4107685809792843317_fin_o,B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3445_Inf__fin_Ounion,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
         => ( ( B != bot_bot_set_set_nat )
           => ( lattic3014633134055518761et_nat(sup_sup_set_set_nat(A2,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,lattic3014633134055518761et_nat(A2)),lattic3014633134055518761et_nat(B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3446_Inf__fin_Ounion,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_real_o(finite_finite_real,B)
         => ( ( B != bot_bot_set_real )
           => ( lattic2677971596711400399n_real(sup_sup_set_real(A2,B)) = aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,lattic2677971596711400399n_real(A2)),lattic2677971596711400399n_real(B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3447_Inf__fin_Ounion,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
         => ( ( B != bot_bo7653980558646680370d_enat )
           => ( lattic974744108425517955d_enat(sup_su4489774667511045786d_enat(A2,B)) = inf_in1870772243966228564d_enat(lattic974744108425517955d_enat(A2),lattic974744108425517955d_enat(B)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3448_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% pos_int_cases
tff(fact_3449_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_3450_reals__Archimedean2,axiom,
    ! [Xa: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% reals_Archimedean2
tff(fact_3451_reals__Archimedean2,axiom,
    ! [Xa: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% reals_Archimedean2
tff(fact_3452_real__arch__simple,axiom,
    ! [Xa: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% real_arch_simple
tff(fact_3453_real__arch__simple,axiom,
    ! [Xa: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% real_arch_simple
tff(fact_3454_finite__enumerate__mono__iff,axiom,
    ! [S: set_Product_unit,M2: nat,Na: nat] :
      ( finite4290736615968046902t_unit(S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),finite410649719033368117t_unit(S))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite410649719033368117t_unit(S))
         => ( ord_le361264281704409273t_unit(infini7930543730640340914t_unit(S,M2),infini7930543730640340914t_unit(S,Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_3455_finite__enumerate__mono__iff,axiom,
    ! [S: set_Extended_enat,M2: nat,Na: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),finite121521170596916366d_enat(S))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite121521170596916366d_enat(S))
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M2)),infini7641415182203889163d_enat(S,Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_3456_finite__enumerate__mono__iff,axiom,
    ! [S: set_nat,M2: nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),finite_card_nat(S))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite_card_nat(S))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M2)),aa_nat_nat(infini8530281810654367211te_nat(S),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_3457_finite__enum__subset,axiom,
    ! [X3: set_Product_unit,Y3: set_Product_unit] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),finite410649719033368117t_unit(X3))
         => ( infini7930543730640340914t_unit(X3,I2) = infini7930543730640340914t_unit(Y3,I2) ) )
     => ( finite4290736615968046902t_unit(X3)
       => ( finite4290736615968046902t_unit(Y3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(X3)),finite410649719033368117t_unit(Y3))
           => ord_le3507040750410214029t_unit(X3,Y3) ) ) ) ) ).

% finite_enum_subset
tff(fact_3458_finite__enum__subset,axiom,
    ! [X3: set_Extended_enat,Y3: set_Extended_enat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),finite121521170596916366d_enat(X3))
         => ( infini7641415182203889163d_enat(X3,I2) = infini7641415182203889163d_enat(Y3,I2) ) )
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,X3)
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,Y3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(X3)),finite121521170596916366d_enat(Y3))
           => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(X3),Y3) ) ) ) ) ).

% finite_enum_subset
tff(fact_3459_finite__enum__subset,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),finite_card_nat(X3))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X3),I2) = aa_nat_nat(infini8530281810654367211te_nat(Y3),I2) ) )
     => ( aa_set_nat_o(finite_finite_nat,X3)
       => ( aa_set_nat_o(finite_finite_nat,Y3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(X3)),finite_card_nat(Y3))
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Y3) ) ) ) ) ).

% finite_enum_subset
tff(fact_3460_card_Oremove,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(Xa,A2)
       => ( finite410649719033368117t_unit(A2) = aa_nat_nat(suc,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)))) ) ) ) ).

% card.remove
tff(fact_3461_card_Oremove,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( member_list_nat(Xa,A2)
       => ( finite_card_list_nat(A2) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)))) ) ) ) ).

% card.remove
tff(fact_3462_card_Oremove,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( finite_card_complex(A2) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% card.remove
tff(fact_3463_card_Oremove,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( finite_card_int(A2) = aa_nat_nat(suc,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)))) ) ) ) ).

% card.remove
tff(fact_3464_card_Oremove,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( finite_card_o(A2) = aa_nat_nat(suc,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)))) ) ) ) ).

% card.remove
tff(fact_3465_card_Oremove,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( finite_card_set_nat(A2) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)))) ) ) ) ).

% card.remove
tff(fact_3466_card_Oremove,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( finite_card_real(A2) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ) ).

% card.remove
tff(fact_3467_card_Oremove,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( finite121521170596916366d_enat(A2) = aa_nat_nat(suc,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% card.remove
tff(fact_3468_card_Oremove,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( finite_card_nat(A2) = aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)))) ) ) ) ).

% card.remove
tff(fact_3469_card_Oinsert__remove,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( finite410649719033368117t_unit(insert_Product_unit(Xa,A2)) = aa_nat_nat(suc,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)))) ) ) ).

% card.insert_remove
tff(fact_3470_card_Oinsert__remove,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite_card_list_nat(insert_list_nat(Xa,A2)) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)))) ) ) ).

% card.insert_remove
tff(fact_3471_card_Oinsert__remove,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite_card_complex(insert_complex(Xa,A2)) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% card.insert_remove
tff(fact_3472_card_Oinsert__remove,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_int(insert_int(Xa,A2)) = aa_nat_nat(suc,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)))) ) ) ).

% card.insert_remove
tff(fact_3473_card_Oinsert__remove,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( finite_card_o(insert_o((Xa),A2)) = aa_nat_nat(suc,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)))) ) ) ).

% card.insert_remove
tff(fact_3474_card_Oinsert__remove,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite_card_set_nat(insert_set_nat(Xa,A2)) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_3475_card_Oinsert__remove,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( finite_card_real(insert_real(Xa,A2)) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ).

% card.insert_remove
tff(fact_3476_card_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite121521170596916366d_enat(insert_Extended_enat(Xa,A2)) = aa_nat_nat(suc,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)))) ) ) ).

% card.insert_remove
tff(fact_3477_card_Oinsert__remove,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_nat(insert_nat(Xa,A2)) = aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_3478_card__Suc__Diff1,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(Xa,A2)
       => ( aa_nat_nat(suc,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A2,insert_Product_unit(Xa,bot_bo3957492148770167129t_unit)))) = finite410649719033368117t_unit(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3479_card__Suc__Diff1,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( member_list_nat(Xa,A2)
       => ( aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(Xa,bot_bot_set_list_nat)))) = finite_card_list_nat(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3480_card__Suc__Diff1,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) = finite_card_complex(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3481_card__Suc__Diff1,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => ( aa_nat_nat(suc,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),insert_int(Xa,bot_bot_set_int)))) = finite_card_int(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3482_card__Suc__Diff1,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( aa_nat_nat(suc,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),insert_o((Xa),bot_bot_set_o)))) = finite_card_o(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3483_card__Suc__Diff1,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat)))) = finite_card_set_nat(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3484_card__Suc__Diff1,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) = finite_card_real(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3485_card__Suc__Diff1,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( aa_nat_nat(suc,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)))) = finite121521170596916366d_enat(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3486_card__Suc__Diff1,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => ( aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat(Xa,bot_bot_set_nat)))) = finite_card_nat(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3487_inverse__of__nat__le,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( ( Na != 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,M2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_3488_inverse__of__nat__le,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( ( Na != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_3489_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y23) )
    <=> ( X23 = Y23 ) ) ).

% nat.inject
tff(fact_3490_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_3491_Int__iff,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),inf_in8357106775501769908d_enat(A2,B))
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
        & aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% Int_iff
tff(fact_3492_Int__iff,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),inf_inf_set_real(A2,B))
    <=> ( aa_set_real_o(member_real(C2),A2)
        & aa_set_real_o(member_real(C2),B) ) ) ).

% Int_iff
tff(fact_3493_Int__iff,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A2,B))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A2)
        & aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% Int_iff
tff(fact_3494_Int__iff,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))
    <=> ( aa_set_nat_o(member_nat(C2),A2)
        & aa_set_nat_o(member_nat(C2),B) ) ) ).

% Int_iff
tff(fact_3495_Int__iff,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))
    <=> ( aa_set_int_o(member_int(C2),A2)
        & aa_set_int_o(member_int(C2),B) ) ) ).

% Int_iff
tff(fact_3496_Int__iff,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B))
    <=> ( aa_set_o_o(member_o((C2)),A2)
        & aa_set_o_o(member_o((C2)),B) ) ) ).

% Int_iff
tff(fact_3497_Int__iff,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))
    <=> ( member8440522571783428010at_nat(C2,A2)
        & member8440522571783428010at_nat(C2,B) ) ) ).

% Int_iff
tff(fact_3498_IntI,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),B)
       => aa_set3337841544853333262enat_o(member_Extended_enat(C2),inf_in8357106775501769908d_enat(A2,B)) ) ) ).

% IntI
tff(fact_3499_IntI,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),A2)
     => ( aa_set_real_o(member_real(C2),B)
       => aa_set_real_o(member_real(C2),inf_inf_set_real(A2,B)) ) ) ).

% IntI
tff(fact_3500_IntI,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A2)
     => ( aa_set_set_nat_o(member_set_nat(C2),B)
       => aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A2,B)) ) ) ).

% IntI
tff(fact_3501_IntI,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A2)
     => ( aa_set_nat_o(member_nat(C2),B)
       => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)) ) ) ).

% IntI
tff(fact_3502_IntI,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),A2)
     => ( aa_set_int_o(member_int(C2),B)
       => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) ) ) ).

% IntI
tff(fact_3503_IntI,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),A2)
     => ( aa_set_o_o(member_o((C2)),B)
       => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)) ) ) ).

% IntI
tff(fact_3504_IntI,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,A2)
     => ( member8440522571783428010at_nat(C2,B)
       => member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) ) ) ).

% IntI
tff(fact_3505_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_3506_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_3507_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_3508_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_3509_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_3510_div__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_3511_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_3512_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_3513_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_3514_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_3515_Suc__less__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% Suc_less_eq
tff(fact_3516_Suc__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_3517_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_3518_Suc__le__mono,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ) ).

% Suc_le_mono
tff(fact_3519_diff__Suc__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) ).

% diff_Suc_Suc
tff(fact_3520_Suc__diff__diff,axiom,
    ! [M2: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),K) ).

% Suc_diff_diff
tff(fact_3521_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% div_self
tff(fact_3522_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_3523_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_3524_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_3525_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_3526_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_3527_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_3528_diff__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),one_one_nat) = Na ).

% diff_Suc_1
tff(fact_3529_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% zle_diff1_eq
tff(fact_3530_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_3531_card__insert__disjoint,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xa),A2)
       => ( finite_card_real(insert_real(Xa,A2)) = aa_nat_nat(suc,finite_card_real(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3532_card__insert__disjoint,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((Xa)),A2)
       => ( finite_card_o(insert_o((Xa),A2)) = aa_nat_nat(suc,finite_card_o(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3533_card__insert__disjoint,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
       => ( finite_card_set_nat(insert_set_nat(Xa,A2)) = aa_nat_nat(suc,finite_card_set_nat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3534_card__insert__disjoint,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ~ member_Product_unit(Xa,A2)
       => ( finite410649719033368117t_unit(insert_Product_unit(Xa,A2)) = aa_nat_nat(suc,finite410649719033368117t_unit(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3535_card__insert__disjoint,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( ~ member_list_nat(Xa,A2)
       => ( finite_card_list_nat(insert_list_nat(Xa,A2)) = aa_nat_nat(suc,finite_card_list_nat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3536_card__insert__disjoint,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),A2)
       => ( finite_card_nat(insert_nat(Xa,A2)) = aa_nat_nat(suc,finite_card_nat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3537_card__insert__disjoint,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xa),A2)
       => ( finite_card_int(insert_int(Xa,A2)) = aa_nat_nat(suc,finite_card_int(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3538_card__insert__disjoint,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xa),A2)
       => ( finite_card_complex(insert_complex(Xa,A2)) = aa_nat_nat(suc,finite_card_complex(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3539_card__insert__disjoint,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => ( finite121521170596916366d_enat(insert_Extended_enat(Xa,A2)) = aa_nat_nat(suc,finite121521170596916366d_enat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3540_enumerate__mono__iff,axiom,
    ! [S: set_Extended_enat,M2: nat,Na: nat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M2)),infini7641415182203889163d_enat(S,Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% enumerate_mono_iff
tff(fact_3541_enumerate__mono__iff,axiom,
    ! [S: set_nat,M2: nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M2)),aa_nat_nat(infini8530281810654367211te_nat(S),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% enumerate_mono_iff
tff(fact_3542_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_3543_int__ops_I6_J,axiom,
    ! [A: nat,B2: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2))) ).

% int_ops(6)
tff(fact_3544_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_3545_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_3546_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_3547_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N2))
           => aa_nat_o(P,N2) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_3548_Int__left__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C)) ).

% Int_left_commute
tff(fact_3549_Int__left__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) ).

% Int_left_absorb
tff(fact_3550_Int__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A2) ).

% Int_commute
tff(fact_3551_Int__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),A2) = A2 ).

% Int_absorb
tff(fact_3552_Int__assoc,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ).

% Int_assoc
tff(fact_3553_IntD2,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),inf_in8357106775501769908d_enat(A2,B))
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ).

% IntD2
tff(fact_3554_IntD2,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),inf_inf_set_real(A2,B))
     => aa_set_real_o(member_real(C2),B) ) ).

% IntD2
tff(fact_3555_IntD2,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A2,B))
     => aa_set_set_nat_o(member_set_nat(C2),B) ) ).

% IntD2
tff(fact_3556_IntD2,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))
     => aa_set_nat_o(member_nat(C2),B) ) ).

% IntD2
tff(fact_3557_IntD2,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))
     => aa_set_int_o(member_int(C2),B) ) ).

% IntD2
tff(fact_3558_IntD2,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B))
     => aa_set_o_o(member_o((C2)),B) ) ).

% IntD2
tff(fact_3559_IntD2,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))
     => member8440522571783428010at_nat(C2,B) ) ).

% IntD2
tff(fact_3560_IntD1,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),inf_in8357106775501769908d_enat(A2,B))
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2) ) ).

% IntD1
tff(fact_3561_IntD1,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),inf_inf_set_real(A2,B))
     => aa_set_real_o(member_real(C2),A2) ) ).

% IntD1
tff(fact_3562_IntD1,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A2,B))
     => aa_set_set_nat_o(member_set_nat(C2),A2) ) ).

% IntD1
tff(fact_3563_IntD1,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))
     => aa_set_nat_o(member_nat(C2),A2) ) ).

% IntD1
tff(fact_3564_IntD1,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))
     => aa_set_int_o(member_int(C2),A2) ) ).

% IntD1
tff(fact_3565_IntD1,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B))
     => aa_set_o_o(member_o((C2)),A2) ) ).

% IntD1
tff(fact_3566_IntD1,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))
     => member8440522571783428010at_nat(C2,A2) ) ).

% IntD1
tff(fact_3567_IntE,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),inf_in8357106775501769908d_enat(A2,B))
     => ~ ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
         => ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),B) ) ) ).

% IntE
tff(fact_3568_IntE,axiom,
    ! [C2: real,A2: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),inf_inf_set_real(A2,B))
     => ~ ( aa_set_real_o(member_real(C2),A2)
         => ~ aa_set_real_o(member_real(C2),B) ) ) ).

% IntE
tff(fact_3569_IntE,axiom,
    ! [C2: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A2,B))
     => ~ ( aa_set_set_nat_o(member_set_nat(C2),A2)
         => ~ aa_set_set_nat_o(member_set_nat(C2),B) ) ) ).

% IntE
tff(fact_3570_IntE,axiom,
    ! [C2: nat,A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))
     => ~ ( aa_set_nat_o(member_nat(C2),A2)
         => ~ aa_set_nat_o(member_nat(C2),B) ) ) ).

% IntE
tff(fact_3571_IntE,axiom,
    ! [C2: int,A2: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))
     => ~ ( aa_set_int_o(member_int(C2),A2)
         => ~ aa_set_int_o(member_int(C2),B) ) ) ).

% IntE
tff(fact_3572_IntE,axiom,
    ! [C2: $o,A2: set_o,B: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B))
     => ~ ( aa_set_o_o(member_o((C2)),A2)
         => ~ aa_set_o_o(member_o((C2)),B) ) ) ).

% IntE
tff(fact_3573_IntE,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))
     => ~ ( member8440522571783428010at_nat(C2,A2)
         => ~ member8440522571783428010at_nat(C2,B) ) ) ).

% IntE
tff(fact_3574_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_3575_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_3576_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_3577_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_3578_verit__la__generic,axiom,
    ! [A: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
      | ( A = Xa )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A) ) ).

% verit_la_generic
tff(fact_3579_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_3580_Suc__inject,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( aa_nat_nat(suc,Xa) = aa_nat_nat(suc,Ya) )
     => ( Xa = Ya ) ) ).

% Suc_inject
tff(fact_3581_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa_nat_nat(suc,Na) ).

% n_not_Suc_n
tff(fact_3582_imp__le__cong,axiom,
    ! [Xa: int,X5: int,P: $o,P2: $o] :
      ( ( Xa = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_3583_conj__le__cong,axiom,
    ! [Xa: int,X5: int,P: $o,P2: $o] :
      ( ( Xa = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_3584_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N2: nat] :
            ( ~ aa_nat_o(P,N2)
            & aa_nat_o(P,aa_nat_nat(suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_3585_enumerate__step,axiom,
    ! [S: set_Extended_enat,Na: nat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,Na)),infini7641415182203889163d_enat(S,aa_nat_nat(suc,Na))) ) ).

% enumerate_step
tff(fact_3586_enumerate__step,axiom,
    ! [S: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),Na)),aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,Na))) ) ).

% enumerate_step
tff(fact_3587_vebt__buildup_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ( ( Xa != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : Xa != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_3588_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M4: nat] : Na = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_3589_Zero__not__Suc,axiom,
    ! [M2: nat] : zero_zero_nat != aa_nat_nat(suc,M2) ).

% Zero_not_Suc
tff(fact_3590_Zero__neq__Suc,axiom,
    ! [M2: nat] : zero_zero_nat != aa_nat_nat(suc,M2) ).

% Zero_neq_Suc
tff(fact_3591_Suc__neq__Zero,axiom,
    ! [M2: nat] : aa_nat_nat(suc,M2) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_3592_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N2))
           => aa_nat_o(P,N2) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_3593_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M2: nat,Na: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y))
       => ( ! [X2: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M2),Na) ) ) ) ).

% diff_induct
tff(fact_3594_nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,N2)
           => aa_nat_o(P,aa_nat_nat(suc,N2)) )
       => aa_nat_o(P,Na) ) ) ).

% nat_induct
tff(fact_3595_old_Onat_Oexhaust,axiom,
    ! [Ya: nat] :
      ( ( Ya != zero_zero_nat )
     => ~ ! [Nat3: nat] : Ya != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_3596_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa_nat_nat(suc,X23) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_3597_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_3598_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_3599_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).

% nat.distinct(1)
tff(fact_3600_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M2))
      <=> ( Na = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_3601_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_3602_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_3603_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_3604_Suc__less__SucD,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% Suc_less_SucD
tff(fact_3605_less__antisym,axiom,
    ! [Na: nat,M2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M2))
       => ( M2 = Na ) ) ) ).

% less_antisym
tff(fact_3606_Suc__less__eq2,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M2)
    <=> ? [M6: nat] :
          ( ( M2 = aa_nat_nat(suc,M6) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M6) ) ) ).

% Suc_less_eq2
tff(fact_3607_All__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Na)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
           => aa_nat_o(P,I4) ) ) ) ).

% All_less_Suc
tff(fact_3608_not__less__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M2)) ) ).

% not_less_eq
tff(fact_3609_less__Suc__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | ( M2 = Na ) ) ) ).

% less_Suc_eq
tff(fact_3610_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Na)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
            & aa_nat_o(P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_3611_less__SucI,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_3612_less__SucE,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => ( M2 = Na ) ) ) ).

% less_SucE
tff(fact_3613_Suc__lessI,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( ( aa_nat_nat(suc,M2) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M2)),Na) ) ) ).

% Suc_lessI
tff(fact_3614_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_3615_Suc__lessD,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M2)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% Suc_lessD
tff(fact_3616_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_3617_diff__less__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),aa_nat_nat(suc,M2)) ).

% diff_less_Suc
tff(fact_3618_Suc__diff__Suc,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) ) ) ).

% Suc_diff_Suc
tff(fact_3619_transitive__stepwise__le,axiom,
    ! [M2: nat,Na: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z2) ) )
         => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N2),aa_nat_nat(suc,N2))
           => aa_nat_o(aa_nat_fun_nat_o(R,M2),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_3620_nat__induct__at__least,axiom,
    ! [M2: nat,Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_o(P,M2)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N2)
             => ( aa_nat_o(P,N2)
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_3621_full__nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M3)),N2)
             => aa_nat_o(P,M3) )
         => aa_nat_o(P,N2) )
     => aa_nat_o(P,Na) ) ).

% full_nat_induct
tff(fact_3622_not__less__eq__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M2) ) ).

% not_less_eq_eq
tff(fact_3623_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_3624_le__Suc__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
        | ( M2 = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_3625_Suc__le__D,axiom,
    ! [Na: nat,M7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M7)
     => ? [M4: nat] : M7 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_3626_le__SucI,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_3627_le__SucE,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( M2 = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_3628_Suc__leD,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% Suc_leD
tff(fact_3629_Suc__diff__le,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) ) ) ).

% Suc_diff_le
tff(fact_3630_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_3631_enumerate__in__set,axiom,
    ! [S: set_Extended_enat,Na: nat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => aa_set3337841544853333262enat_o(member_Extended_enat(infini7641415182203889163d_enat(S,Na)),S) ) ).

% enumerate_in_set
tff(fact_3632_enumerate__in__set,axiom,
    ! [S: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_set_nat_o(member_nat(aa_nat_nat(infini8530281810654367211te_nat(S),Na)),S) ) ).

% enumerate_in_set
tff(fact_3633_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
     => ~ ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E2) ) ).

% nat_approx_posE
tff(fact_3634_nat__approx__posE,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ~ ! [N2: 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,N2)))),E2) ) ).

% nat_approx_posE
tff(fact_3635_enumerate__Ex,axiom,
    ! [S: set_nat,S3: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(member_nat(S3),S)
       => ? [N2: nat] : aa_nat_nat(infini8530281810654367211te_nat(S),N2) = S3 ) ) ).

% enumerate_Ex
tff(fact_3636_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Na: nat,M2: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,M2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3637_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Na: nat,M2: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,M2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3638_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Na: nat,M2: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,M2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3639_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Na: nat,M2: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,M2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3640_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Na: nat,M2: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,M2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3641_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N5)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_3642_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_3643_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N5)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_3644_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_3645_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N5)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_3646_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_3647_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_3648_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_3649_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_3650_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N5)),aa_nat_set_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_3651_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N5)),aa_nat_rat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_3652_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N5)),aa_nat_num(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_3653_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N5)),aa_nat_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_3654_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N5)),aa_nat_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_3655_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Na)),aa_nat_set_int(F,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_3656_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_3657_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Na)),aa_nat_num(F,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_3658_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_3659_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N5: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Na)),aa_nat_int(F,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_3660_finite__enumerate__step,axiom,
    ! [S: set_Product_unit,Na: nat] :
      ( finite4290736615968046902t_unit(S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),finite410649719033368117t_unit(S))
       => ord_le361264281704409273t_unit(infini7930543730640340914t_unit(S,Na),infini7930543730640340914t_unit(S,aa_nat_nat(suc,Na))) ) ) ).

% finite_enumerate_step
tff(fact_3661_finite__enumerate__step,axiom,
    ! [S: set_Extended_enat,Na: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),finite121521170596916366d_enat(S))
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,Na)),infini7641415182203889163d_enat(S,aa_nat_nat(suc,Na))) ) ) ).

% finite_enumerate_step
tff(fact_3662_finite__enumerate__step,axiom,
    ! [S: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),finite_card_nat(S))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),Na)),aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,Na))) ) ) ).

% finite_enumerate_step
tff(fact_3663_enumerate__Suc_H,axiom,
    ! [S: set_Extended_enat,Na: nat] : infini7641415182203889163d_enat(S,aa_nat_nat(suc,Na)) = infini7641415182203889163d_enat(minus_925952699566721837d_enat(S,insert_Extended_enat(infini7641415182203889163d_enat(S,zero_zero_nat),bot_bo7653980558646680370d_enat)),Na) ).

% enumerate_Suc'
tff(fact_3664_enumerate__Suc_H,axiom,
    ! [S: set_nat,Na: nat] : aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,Na)) = aa_nat_nat(infini8530281810654367211te_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(aa_nat_nat(infini8530281810654367211te_nat(S),zero_zero_nat),bot_bot_set_nat))),Na) ).

% enumerate_Suc'
tff(fact_3665_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(suc,Na))
    <=> ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2 = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_3666_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M4: nat] : Na = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_3667_All__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_3668_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M: nat] : Na = aa_nat_nat(suc,M) ) ).

% gr0_conv_Suc
tff(fact_3669_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
            & aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_3670_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_3671_le__imp__less__Suc,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_3672_less__eq__Suc__le,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M2) ) ).

% less_eq_Suc_le
tff(fact_3673_less__Suc__eq__le,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% less_Suc_eq_le
tff(fact_3674_le__less__Suc__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M2))
      <=> ( Na = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_3675_Suc__le__lessD,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% Suc_le_lessD
tff(fact_3676_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N2))
                 => aa_nat_o(P,N2) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_3677_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P,N2)
                 => aa_nat_o(P,aa_nat_nat(suc,N2)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_3678_Suc__le__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% Suc_le_eq
tff(fact_3679_Suc__leI,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),Na) ) ).

% Suc_leI
tff(fact_3680_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3681_le__enumerate,axiom,
    ! [S: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(infini8530281810654367211te_nat(S),Na)) ) ).

% le_enumerate
tff(fact_3682_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K2)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_3683_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(P,N2)
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_3684_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3685_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3686_card__insert__if,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( finite_card_real(insert_real(Xa,A2)) = $ite(aa_set_real_o(member_real(Xa),A2),finite_card_real(A2),aa_nat_nat(suc,finite_card_real(A2))) ) ) ).

% card_insert_if
tff(fact_3687_card__insert__if,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( finite_card_o(insert_o((Xa),A2)) = $ite(aa_set_o_o(member_o((Xa)),A2),finite_card_o(A2),aa_nat_nat(suc,finite_card_o(A2))) ) ) ).

% card_insert_if
tff(fact_3688_card__insert__if,axiom,
    ! [A2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite_card_set_nat(insert_set_nat(Xa,A2)) = $ite(aa_set_set_nat_o(member_set_nat(Xa),A2),finite_card_set_nat(A2),aa_nat_nat(suc,finite_card_set_nat(A2))) ) ) ).

% card_insert_if
tff(fact_3689_card__insert__if,axiom,
    ! [A2: set_Product_unit,Xa: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( finite410649719033368117t_unit(insert_Product_unit(Xa,A2)) = $ite(member_Product_unit(Xa,A2),finite410649719033368117t_unit(A2),aa_nat_nat(suc,finite410649719033368117t_unit(A2))) ) ) ).

% card_insert_if
tff(fact_3690_card__insert__if,axiom,
    ! [A2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite_card_list_nat(insert_list_nat(Xa,A2)) = $ite(member_list_nat(Xa,A2),finite_card_list_nat(A2),aa_nat_nat(suc,finite_card_list_nat(A2))) ) ) ).

% card_insert_if
tff(fact_3691_card__insert__if,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_nat(insert_nat(Xa,A2)) = $ite(aa_set_nat_o(member_nat(Xa),A2),finite_card_nat(A2),aa_nat_nat(suc,finite_card_nat(A2))) ) ) ).

% card_insert_if
tff(fact_3692_card__insert__if,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_int(insert_int(Xa,A2)) = $ite(aa_set_int_o(member_int(Xa),A2),finite_card_int(A2),aa_nat_nat(suc,finite_card_int(A2))) ) ) ).

% card_insert_if
tff(fact_3693_card__insert__if,axiom,
    ! [A2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite_card_complex(insert_complex(Xa,A2)) = $ite(aa_set_complex_o(member_complex(Xa),A2),finite_card_complex(A2),aa_nat_nat(suc,finite_card_complex(A2))) ) ) ).

% card_insert_if
tff(fact_3694_card__insert__if,axiom,
    ! [A2: set_Extended_enat,Xa: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite121521170596916366d_enat(insert_Extended_enat(Xa,A2)) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2),finite121521170596916366d_enat(A2),aa_nat_nat(suc,finite121521170596916366d_enat(A2))) ) ) ).

% card_insert_if
tff(fact_3695_card__Suc__eq__finite,axiom,
    ! [A2: set_real,K: nat] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: real,B10: set_real] :
          ( ( A2 = insert_real(B6,B10) )
          & ~ aa_set_real_o(member_real(B6),B10)
          & ( finite_card_real(B10) = K )
          & aa_set_real_o(finite_finite_real,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3696_card__Suc__eq__finite,axiom,
    ! [A2: set_o,K: nat] :
      ( ( finite_card_o(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: $o,B10: set_o] :
          ( ( A2 = insert_o((B6),B10) )
          & ~ aa_set_o_o(member_o((B6)),B10)
          & ( finite_card_o(B10) = K )
          & aa_set_o_o(finite_finite_o,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3697_card__Suc__eq__finite,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: set_nat,B10: set_set_nat] :
          ( ( A2 = insert_set_nat(B6,B10) )
          & ~ aa_set_set_nat_o(member_set_nat(B6),B10)
          & ( finite_card_set_nat(B10) = K )
          & aa_set_set_nat_o(finite1152437895449049373et_nat,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3698_card__Suc__eq__finite,axiom,
    ! [A2: set_Product_unit,K: nat] :
      ( ( finite410649719033368117t_unit(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: product_unit,B10: set_Product_unit] :
          ( ( A2 = insert_Product_unit(B6,B10) )
          & ~ member_Product_unit(B6,B10)
          & ( finite410649719033368117t_unit(B10) = K )
          & finite4290736615968046902t_unit(B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3699_card__Suc__eq__finite,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: list_nat,B10: set_list_nat] :
          ( ( A2 = insert_list_nat(B6,B10) )
          & ~ member_list_nat(B6,B10)
          & ( finite_card_list_nat(B10) = K )
          & finite8100373058378681591st_nat(B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3700_card__Suc__eq__finite,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( finite_card_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: nat,B10: set_nat] :
          ( ( A2 = insert_nat(B6,B10) )
          & ~ aa_set_nat_o(member_nat(B6),B10)
          & ( finite_card_nat(B10) = K )
          & aa_set_nat_o(finite_finite_nat,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3701_card__Suc__eq__finite,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: int,B10: set_int] :
          ( ( A2 = insert_int(B6,B10) )
          & ~ aa_set_int_o(member_int(B6),B10)
          & ( finite_card_int(B10) = K )
          & aa_set_int_o(finite_finite_int,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3702_card__Suc__eq__finite,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: complex,B10: set_complex] :
          ( ( A2 = insert_complex(B6,B10) )
          & ~ aa_set_complex_o(member_complex(B6),B10)
          & ( finite_card_complex(B10) = K )
          & aa_set_complex_o(finite3207457112153483333omplex,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3703_card__Suc__eq__finite,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite121521170596916366d_enat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: extended_enat,B10: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(B6,B10) )
          & ~ aa_set3337841544853333262enat_o(member_Extended_enat(B6),B10)
          & ( finite121521170596916366d_enat(B10) = K )
          & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3704_zle__int,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% zle_int
tff(fact_3705_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B2: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B2))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_3706_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_3707_enumerate__mono,axiom,
    ! [M2: nat,Na: nat,S: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M2)),infini7641415182203889163d_enat(S,Na)) ) ) ).

% enumerate_mono
tff(fact_3708_enumerate__mono,axiom,
    ! [M2: nat,Na: nat,S: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M2)),aa_nat_nat(infini8530281810654367211te_nat(S),Na)) ) ) ).

% enumerate_mono
tff(fact_3709_finite__le__enumerate,axiom,
    ! [S: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite_card_nat(S))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(infini8530281810654367211te_nat(S),Na)) ) ) ).

% finite_le_enumerate
tff(fact_3710_finite__enumerate__in__set,axiom,
    ! [S: set_Product_unit,Na: nat] :
      ( finite4290736615968046902t_unit(S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite410649719033368117t_unit(S))
       => member_Product_unit(infini7930543730640340914t_unit(S,Na),S) ) ) ).

% finite_enumerate_in_set
tff(fact_3711_finite__enumerate__in__set,axiom,
    ! [S: set_Extended_enat,Na: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite121521170596916366d_enat(S))
       => aa_set3337841544853333262enat_o(member_Extended_enat(infini7641415182203889163d_enat(S,Na)),S) ) ) ).

% finite_enumerate_in_set
tff(fact_3712_finite__enumerate__in__set,axiom,
    ! [S: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite_card_nat(S))
       => aa_set_nat_o(member_nat(aa_nat_nat(infini8530281810654367211te_nat(S),Na)),S) ) ) ).

% finite_enumerate_in_set
tff(fact_3713_finite__enumerate__Ex,axiom,
    ! [S: set_Product_unit,S3: product_unit] :
      ( finite4290736615968046902t_unit(S)
     => ( member_Product_unit(S3,S)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),finite410649719033368117t_unit(S))
            & ( infini7930543730640340914t_unit(S,N2) = S3 ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_3714_finite__enumerate__Ex,axiom,
    ! [S: set_Extended_enat,S3: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(S3),S)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),finite121521170596916366d_enat(S))
            & ( infini7641415182203889163d_enat(S,N2) = S3 ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_3715_finite__enumerate__Ex,axiom,
    ! [S: set_nat,S3: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(member_nat(S3),S)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),finite_card_nat(S))
            & ( aa_nat_nat(infini8530281810654367211te_nat(S),N2) = S3 ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_3716_finite__enum__ext,axiom,
    ! [X3: set_Product_unit,Y3: set_Product_unit] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),finite410649719033368117t_unit(X3))
         => ( infini7930543730640340914t_unit(X3,I2) = infini7930543730640340914t_unit(Y3,I2) ) )
     => ( finite4290736615968046902t_unit(X3)
       => ( finite4290736615968046902t_unit(Y3)
         => ( ( finite410649719033368117t_unit(X3) = finite410649719033368117t_unit(Y3) )
           => ( X3 = Y3 ) ) ) ) ) ).

% finite_enum_ext
tff(fact_3717_finite__enum__ext,axiom,
    ! [X3: set_Extended_enat,Y3: set_Extended_enat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),finite121521170596916366d_enat(X3))
         => ( infini7641415182203889163d_enat(X3,I2) = infini7641415182203889163d_enat(Y3,I2) ) )
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,X3)
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,Y3)
         => ( ( finite121521170596916366d_enat(X3) = finite121521170596916366d_enat(Y3) )
           => ( X3 = Y3 ) ) ) ) ) ).

% finite_enum_ext
tff(fact_3718_finite__enum__ext,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),finite_card_nat(X3))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X3),I2) = aa_nat_nat(infini8530281810654367211te_nat(Y3),I2) ) )
     => ( aa_set_nat_o(finite_finite_nat,X3)
       => ( aa_set_nat_o(finite_finite_nat,Y3)
         => ( ( finite_card_nat(X3) = finite_card_nat(Y3) )
           => ( X3 = Y3 ) ) ) ) ) ).

% finite_enum_ext
tff(fact_3719_card__Suc__eq,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: complex,B10: set_complex] :
          ( ( A2 = insert_complex(B6,B10) )
          & ~ aa_set_complex_o(member_complex(B6),B10)
          & ( finite_card_complex(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_complex ) ) ) ) ).

% card_Suc_eq
tff(fact_3720_card__Suc__eq,axiom,
    ! [A2: set_Product_unit,K: nat] :
      ( ( finite410649719033368117t_unit(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: product_unit,B10: set_Product_unit] :
          ( ( A2 = insert_Product_unit(B6,B10) )
          & ~ member_Product_unit(B6,B10)
          & ( finite410649719033368117t_unit(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bo3957492148770167129t_unit ) ) ) ) ).

% card_Suc_eq
tff(fact_3721_card__Suc__eq,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: list_nat,B10: set_list_nat] :
          ( ( A2 = insert_list_nat(B6,B10) )
          & ~ member_list_nat(B6,B10)
          & ( finite_card_list_nat(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_list_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_3722_card__Suc__eq,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( finite_card_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: nat,B10: set_nat] :
          ( ( A2 = insert_nat(B6,B10) )
          & ~ aa_set_nat_o(member_nat(B6),B10)
          & ( finite_card_nat(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_3723_card__Suc__eq,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: int,B10: set_int] :
          ( ( A2 = insert_int(B6,B10) )
          & ~ aa_set_int_o(member_int(B6),B10)
          & ( finite_card_int(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_int ) ) ) ) ).

% card_Suc_eq
tff(fact_3724_card__Suc__eq,axiom,
    ! [A2: set_o,K: nat] :
      ( ( finite_card_o(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: $o,B10: set_o] :
          ( ( A2 = insert_o((B6),B10) )
          & ~ aa_set_o_o(member_o((B6)),B10)
          & ( finite_card_o(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_o ) ) ) ) ).

% card_Suc_eq
tff(fact_3725_card__Suc__eq,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: set_nat,B10: set_set_nat] :
          ( ( A2 = insert_set_nat(B6,B10) )
          & ~ aa_set_set_nat_o(member_set_nat(B6),B10)
          & ( finite_card_set_nat(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_3726_card__Suc__eq,axiom,
    ! [A2: set_real,K: nat] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: real,B10: set_real] :
          ( ( A2 = insert_real(B6,B10) )
          & ~ aa_set_real_o(member_real(B6),B10)
          & ( finite_card_real(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bot_set_real ) ) ) ) ).

% card_Suc_eq
tff(fact_3727_card__Suc__eq,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite121521170596916366d_enat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B6: extended_enat,B10: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(B6,B10) )
          & ~ aa_set3337841544853333262enat_o(member_Extended_enat(B6),B10)
          & ( finite121521170596916366d_enat(B10) = K )
          & ( ( K = zero_zero_nat )
           => ( B10 = bot_bo7653980558646680370d_enat ) ) ) ) ).

% card_Suc_eq
tff(fact_3728_card__eq__SucD,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,K) )
     => ? [B4: complex,B3: set_complex] :
          ( ( A2 = insert_complex(B4,B3) )
          & ~ aa_set_complex_o(member_complex(B4),B3)
          & ( finite_card_complex(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
tff(fact_3729_card__eq__SucD,axiom,
    ! [A2: set_Product_unit,K: nat] :
      ( ( finite410649719033368117t_unit(A2) = aa_nat_nat(suc,K) )
     => ? [B4: product_unit,B3: set_Product_unit] :
          ( ( A2 = insert_Product_unit(B4,B3) )
          & ~ member_Product_unit(B4,B3)
          & ( finite410649719033368117t_unit(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bo3957492148770167129t_unit ) ) ) ) ).

% card_eq_SucD
tff(fact_3730_card__eq__SucD,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,K) )
     => ? [B4: list_nat,B3: set_list_nat] :
          ( ( A2 = insert_list_nat(B4,B3) )
          & ~ member_list_nat(B4,B3)
          & ( finite_card_list_nat(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_3731_card__eq__SucD,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( finite_card_nat(A2) = aa_nat_nat(suc,K) )
     => ? [B4: nat,B3: set_nat] :
          ( ( A2 = insert_nat(B4,B3) )
          & ~ aa_set_nat_o(member_nat(B4),B3)
          & ( finite_card_nat(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_3732_card__eq__SucD,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,K) )
     => ? [B4: int,B3: set_int] :
          ( ( A2 = insert_int(B4,B3) )
          & ~ aa_set_int_o(member_int(B4),B3)
          & ( finite_card_int(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
tff(fact_3733_card__eq__SucD,axiom,
    ! [A2: set_o,K: nat] :
      ( ( finite_card_o(A2) = aa_nat_nat(suc,K) )
     => ? [B4: $o,B3: set_o] :
          ( ( A2 = insert_o((B4),B3) )
          & ~ aa_set_o_o(member_o((B4)),B3)
          & ( finite_card_o(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
tff(fact_3734_card__eq__SucD,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,K) )
     => ? [B4: set_nat,B3: set_set_nat] :
          ( ( A2 = insert_set_nat(B4,B3) )
          & ~ aa_set_set_nat_o(member_set_nat(B4),B3)
          & ( finite_card_set_nat(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_3735_card__eq__SucD,axiom,
    ! [A2: set_real,K: nat] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,K) )
     => ? [B4: real,B3: set_real] :
          ( ( A2 = insert_real(B4,B3) )
          & ~ aa_set_real_o(member_real(B4),B3)
          & ( finite_card_real(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
tff(fact_3736_card__eq__SucD,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite121521170596916366d_enat(A2) = aa_nat_nat(suc,K) )
     => ? [B4: extended_enat,B3: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(B4,B3) )
          & ~ aa_set3337841544853333262enat_o(member_Extended_enat(B4),B3)
          & ( finite121521170596916366d_enat(B3) = K )
          & ( ( K = zero_zero_nat )
           => ( B3 = bot_bo7653980558646680370d_enat ) ) ) ) ).

% card_eq_SucD
tff(fact_3737_card__1__singleton__iff,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: complex] : A2 = insert_complex(X,bot_bot_set_complex) ) ).

% card_1_singleton_iff
tff(fact_3738_card__1__singleton__iff,axiom,
    ! [A2: set_Product_unit] :
      ( ( finite410649719033368117t_unit(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: product_unit] : A2 = insert_Product_unit(X,bot_bo3957492148770167129t_unit) ) ).

% card_1_singleton_iff
tff(fact_3739_card__1__singleton__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: list_nat] : A2 = insert_list_nat(X,bot_bot_set_list_nat) ) ).

% card_1_singleton_iff
tff(fact_3740_card__1__singleton__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_card_nat(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: nat] : A2 = insert_nat(X,bot_bot_set_nat) ) ).

% card_1_singleton_iff
tff(fact_3741_card__1__singleton__iff,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: int] : A2 = insert_int(X,bot_bot_set_int) ) ).

% card_1_singleton_iff
tff(fact_3742_card__1__singleton__iff,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: $o] : A2 = insert_o((X),bot_bot_set_o) ) ).

% card_1_singleton_iff
tff(fact_3743_card__1__singleton__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: set_nat] : A2 = insert_set_nat(X,bot_bot_set_set_nat) ) ).

% card_1_singleton_iff
tff(fact_3744_card__1__singleton__iff,axiom,
    ! [A2: set_real] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: real] : A2 = insert_real(X,bot_bot_set_real) ) ).

% card_1_singleton_iff
tff(fact_3745_card__1__singleton__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X: extended_enat] : A2 = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ).

% card_1_singleton_iff
tff(fact_3746_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X),A2)
           => ! [Xa3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa3),A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3747_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: product_unit] :
            ( member_Product_unit(X,A2)
           => ! [Xa3: product_unit] :
                ( member_Product_unit(Xa3,A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3748_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: list_nat] :
            ( member_list_nat(X,A2)
           => ! [Xa3: list_nat] :
                ( member_list_nat(Xa3,A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3749_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
           => ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3750_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: int] :
            ( aa_set_int_o(member_int(X),A2)
           => ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3751_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
           => ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3752_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
           => ! [Xa3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa3),A2)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3753_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_real(A2))
    <=> ? [A7: real,B10: set_real] :
          ( ( A2 = insert_real(A7,B10) )
          & ~ aa_set_real_o(member_real(A7),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_real(B10))
          & aa_set_real_o(finite_finite_real,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3754_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_o(A2))
    <=> ? [A7: $o,B10: set_o] :
          ( ( A2 = insert_o((A7),B10) )
          & ~ aa_set_o_o(member_o((A7)),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_o(B10))
          & aa_set_o_o(finite_finite_o,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3755_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_set_nat(A2))
    <=> ? [A7: set_nat,B10: set_set_nat] :
          ( ( A2 = insert_set_nat(A7,B10) )
          & ~ aa_set_set_nat_o(member_set_nat(A7),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_set_nat(B10))
          & aa_set_set_nat_o(finite1152437895449049373et_nat,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3756_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite410649719033368117t_unit(A2))
    <=> ? [A7: product_unit,B10: set_Product_unit] :
          ( ( A2 = insert_Product_unit(A7,B10) )
          & ~ member_Product_unit(A7,B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite410649719033368117t_unit(B10))
          & finite4290736615968046902t_unit(B10) ) ) ).

% card_le_Suc_iff
tff(fact_3757_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_list_nat(A2))
    <=> ? [A7: list_nat,B10: set_list_nat] :
          ( ( A2 = insert_list_nat(A7,B10) )
          & ~ member_list_nat(A7,B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_list_nat(B10))
          & finite8100373058378681591st_nat(B10) ) ) ).

% card_le_Suc_iff
tff(fact_3758_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_nat(A2))
    <=> ? [A7: nat,B10: set_nat] :
          ( ( A2 = insert_nat(A7,B10) )
          & ~ aa_set_nat_o(member_nat(A7),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_nat(B10))
          & aa_set_nat_o(finite_finite_nat,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3759_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_int(A2))
    <=> ? [A7: int,B10: set_int] :
          ( ( A2 = insert_int(A7,B10) )
          & ~ aa_set_int_o(member_int(A7),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_int(B10))
          & aa_set_int_o(finite_finite_int,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3760_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_complex(A2))
    <=> ? [A7: complex,B10: set_complex] :
          ( ( A2 = insert_complex(A7,B10) )
          & ~ aa_set_complex_o(member_complex(A7),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_complex(B10))
          & aa_set_complex_o(finite3207457112153483333omplex,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3761_card__le__Suc__iff,axiom,
    ! [Na: nat,A2: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite121521170596916366d_enat(A2))
    <=> ? [A7: extended_enat,B10: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(A7,B10) )
          & ~ aa_set3337841544853333262enat_o(member_Extended_enat(A7),B10)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite121521170596916366d_enat(B10))
          & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B10) ) ) ).

% card_le_Suc_iff
tff(fact_3762_finite__enumerate__mono,axiom,
    ! [M2: nat,Na: nat,S: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( finite4290736615968046902t_unit(S)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite410649719033368117t_unit(S))
         => ord_le361264281704409273t_unit(infini7930543730640340914t_unit(S,M2),infini7930543730640340914t_unit(S,Na)) ) ) ) ).

% finite_enumerate_mono
tff(fact_3763_finite__enumerate__mono,axiom,
    ! [M2: nat,Na: nat,S: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite121521170596916366d_enat(S))
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M2)),infini7641415182203889163d_enat(S,Na)) ) ) ) ).

% finite_enumerate_mono
tff(fact_3764_finite__enumerate__mono,axiom,
    ! [M2: nat,Na: nat,S: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_set_nat_o(finite_finite_nat,S)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),finite_card_nat(S))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M2)),aa_nat_nat(infini8530281810654367211te_nat(S),Na)) ) ) ) ).

% finite_enumerate_mono
tff(fact_3765_divide__le__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_3766_divide__le__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_3767_divide__le__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_3768_divide__le__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_3769_le__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_3770_le__divide__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_3771_le__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_3772_le__divide__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_3773_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_3774_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_3775_less__divide__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_3776_less__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_3777_less__divide__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_3778_less__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_3779_divide__less__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_3780_divide__less__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_3781_divide__less__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_3782_divide__less__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_3783_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_3784_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_3785_divide__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_3786_divide__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_3787_divide__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_3788_divide__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_3789_divide__cancel__left,axiom,
    ! [C2: real,A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_3790_divide__cancel__left,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C2),A) = aa_complex_complex(divide1717551699836669952omplex(C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_3791_divide__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_3792_divide__cancel__right,axiom,
    ! [A: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_3793_divide__cancel__right,axiom,
    ! [A: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C2) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_3794_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_3795_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_3796_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_3797_divide__eq__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) = one_one_rat )
    <=> ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_3798_divide__eq__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = one_one_real )
    <=> ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_3799_divide__eq__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = one_one_complex )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_3800_one__eq__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_3801_one__eq__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) )
    <=> ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_3802_one__eq__divide__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B2) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_3803_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_3804_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_3805_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_3806_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_3807_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_3808_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_3809_divide__eq__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_3810_divide__eq__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_3811_eq__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_3812_eq__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_3813_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_3814_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_3815_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_3816_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_3817_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_3818_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_3819_divide__le__0__1__iff,axiom,
    ! [A: 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),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_3820_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_3821_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_3822_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_3823_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X4),X_1) ).

% linordered_field_no_ub
tff(fact_3824_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),X_1) ).

% linordered_field_no_ub
tff(fact_3825_divide__right__mono__neg,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_3826_divide__right__mono__neg,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_3827_divide__nonpos__nonpos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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,Xa),Ya)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3828_divide__nonpos__nonpos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3829_divide__nonpos__nonneg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_3830_divide__nonpos__nonneg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3831_divide__nonneg__nonpos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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,Xa),Ya)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_3832_divide__nonneg__nonpos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3833_divide__nonneg__nonneg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa),Ya)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3834_divide__nonneg__nonneg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3835_zero__le__divide__iff,axiom,
    ! [A: 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,A),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_3836_zero__le__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3837_divide__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ) ) ).

% divide_right_mono
tff(fact_3838_divide__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)) ) ) ).

% divide_right_mono
tff(fact_3839_divide__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_3840_divide__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_3841_divide__neg__neg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% divide_neg_neg
tff(fact_3842_divide__neg__neg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,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,Xa),Ya)) ) ) ).

% divide_neg_neg
tff(fact_3843_divide__neg__pos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_3844_divide__neg__pos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_3845_divide__pos__neg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_3846_divide__pos__neg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_3847_divide__pos__pos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% divide_pos_pos
tff(fact_3848_divide__pos__pos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) ) ) ).

% divide_pos_pos
tff(fact_3849_divide__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_3850_divide__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_3851_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),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),A) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3852_divide__less__cancel,axiom,
    ! [A: 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,A),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,A),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),A) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_3853_zero__less__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_3854_zero__less__divide__iff,axiom,
    ! [A: 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,A),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_3855_divide__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_3856_divide__strict__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_3857_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3858_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3859_right__inverse__eq,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) = one_one_rat )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_3860_right__inverse__eq,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = one_one_real )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_3861_right__inverse__eq,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = one_one_complex )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_3862_divide__nonpos__pos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3863_divide__nonpos__pos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3864_divide__nonpos__neg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,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,Xa),Ya)) ) ) ).

% divide_nonpos_neg
tff(fact_3865_divide__nonpos__neg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% divide_nonpos_neg
tff(fact_3866_divide__nonneg__pos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),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,Xa),Ya)) ) ) ).

% divide_nonneg_pos
tff(fact_3867_divide__nonneg__pos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% divide_nonneg_pos
tff(fact_3868_divide__nonneg__neg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,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,Xa),Ya)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3869_divide__nonneg__neg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3870_divide__le__cancel,axiom,
    ! [A: 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,A),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,A),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),A) ) ) ) ).

% divide_le_cancel
tff(fact_3871_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% divide_le_cancel
tff(fact_3872_frac__less2,axiom,
    ! [Xa: real,Ya: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),W2)) ) ) ) ) ).

% frac_less2
tff(fact_3873_frac__less2,axiom,
    ! [Xa: rat,Ya: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ya),W2)) ) ) ) ) ).

% frac_less2
tff(fact_3874_frac__less,axiom,
    ! [Xa: real,Ya: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),W2)) ) ) ) ) ).

% frac_less
tff(fact_3875_frac__less,axiom,
    ! [Xa: rat,Ya: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ya),W2)) ) ) ) ) ).

% frac_less
tff(fact_3876_frac__le,axiom,
    ! [Ya: real,Xa: real,W2: 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,Xa),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( 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,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),W2)) ) ) ) ) ).

% frac_le
tff(fact_3877_frac__le,axiom,
    ! [Ya: rat,Xa: rat,W2: 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,Xa),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( 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,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ya),W2)) ) ) ) ) ).

% frac_le
tff(fact_3878_divide__less__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3879_divide__less__eq__1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3880_less__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3881_less__divide__eq__1,axiom,
    ! [B2: real,A: 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),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3882_le__divide__eq__1,axiom,
    ! [B2: real,A: 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),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3883_le__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3884_divide__le__eq__1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3885_divide__le__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3886_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_3887_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_3888_le__div__geq,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_3889_div__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = zero_zero_nat ) ) ).

% div_less
tff(fact_3890_div__by__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = M2 ).

% div_by_Suc_0
tff(fact_3891_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_3892_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_3893_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_3894_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_3895_real__of__nat__div3,axiom,
    ! [Na: nat,Xa: 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,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa)))),one_one_real) ).

% real_of_nat_div3
tff(fact_3896_real__of__nat__div4,axiom,
    ! [Na: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))) ).

% real_of_nat_div4
tff(fact_3897_real__of__nat__div2,axiom,
    ! [Na: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa)))) ).

% real_of_nat_div2
tff(fact_3898_div__le__dividend,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),M2) ).

% div_le_dividend
tff(fact_3899_div__le__mono,axiom,
    ! [M2: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),K)) ) ).

% div_le_mono
tff(fact_3900_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | ( Na = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3901_Suc__div__le__mono,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M2)),Na)) ).

% Suc_div_le_mono
tff(fact_3902_div__le__mono2,axiom,
    ! [M2: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M2)) ) ) ).

% div_le_mono2
tff(fact_3903_div__greater__zero__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_3904_div__eq__dividend__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = M2 )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3905_div__less__dividend,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),M2) ) ) ).

% div_less_dividend
tff(fact_3906_div__if,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na)) ) ).

% div_if
tff(fact_3907_div__geq,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_3908_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_3909_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3910_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: 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,A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3911_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: 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,A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3912_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_3913_div__nonpos__pos__le0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3914_div__nonneg__neg__le0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,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,A),B2)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3915_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_3916_ln__inj__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(ln_ln_real,Ya) )
        <=> ( Xa = Ya ) ) ) ) ).

% ln_inj_iff
tff(fact_3917_ln__less__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ) ).

% ln_less_cancel_iff
tff(fact_3918_ln__le__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ) ).

% ln_le_cancel_iff
tff(fact_3919_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_3920_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_3921_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = zero_zero_real )
      <=> ( Xa = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_3922_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_3923_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_3924_ln__div,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Ya)) ) ) ) ).

% ln_div
tff(fact_3925_ln__diff__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),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,Xa)),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,Xa),Ya)),Ya)) ) ) ).

% ln_diff_le
tff(fact_3926_ln__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_bound
tff(fact_3927_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_less_self
tff(fact_3928_less__eq__real__def,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% less_eq_real_def
tff(fact_3929_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),S)
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_set_real_o(member_real(X2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),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] :
                ( ! [X2: real] :
                    ( aa_set_real_o(member_real(X2),S)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z3) ) ) ) ) ).

% complete_real
tff(fact_3930_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_3931_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),one_one_real) )
       => ( Xa = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_3932_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_3933_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_gt_zero
tff(fact_3934_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_ge_zero
tff(fact_3935_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_3936_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_3937_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: 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,A),B2)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3938_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: 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,A),B2)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3939_div__neg__pos__less0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_3940_div__positive,axiom,
    ! [B2: nat,A: 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),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_positive
tff(fact_3941_div__positive,axiom,
    ! [B2: int,A: 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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_positive
tff(fact_3942_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3943_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3944_zdiv__mono1,axiom,
    ! [A: int,A9: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A9)
     => ( 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,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A9),B2)) ) ) ).

% zdiv_mono1
tff(fact_3945_zdiv__mono2,axiom,
    ! [A: int,B8: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B8)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B8),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B8)) ) ) ) ).

% zdiv_mono2
tff(fact_3946_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_3947_zdiv__mono1__neg,axiom,
    ! [A: int,A9: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A9)
     => ( 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,A9),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_3948_zdiv__mono2__neg,axiom,
    ! [A: int,B8: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B8)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B8),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B8)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3949_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_3950_Bolzano,axiom,
    ! [A: real,B2: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [A3: real,B4: real,C4: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B4)
           => ( aa_real_o(aa_real_fun_real_o(P,B4),C4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),C4)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C4) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
               => ? [D3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
                    & ! [A3: real,B4: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B4)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A3)),D3) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B4) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B2) ) ) ) ).

% Bolzano
tff(fact_3951_int__power__div__base,axiom,
    ! [M2: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( 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),M2)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3952_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3953_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ~ ! [N2: nat] : Xa != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_3954_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_3955_nat__intermed__int__val,axiom,
    ! [M2: nat,Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M2)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3956_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_3957_add__left__cancel,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_3958_add__left__cancel,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_3959_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_3960_add__left__cancel,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_3961_add__right__cancel,axiom,
    ! [B2: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_3962_add__right__cancel,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_3963_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_3964_add__right__cancel,axiom,
    ! [B2: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_3965_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_3966_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_3967_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_3968_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_3969_add__le__cancel__right,axiom,
    ! [A: 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_cancel_right
tff(fact_3970_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_cancel_right
tff(fact_3971_add__le__cancel__right,axiom,
    ! [A: 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,A),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,A),B2) ) ).

% add_le_cancel_right
tff(fact_3972_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_cancel_right
tff(fact_3973_add__le__cancel__left,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_cancel_left
tff(fact_3974_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_cancel_left
tff(fact_3975_add__le__cancel__left,axiom,
    ! [C2: nat,A: 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),A)),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,A),B2) ) ).

% add_le_cancel_left
tff(fact_3976_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_cancel_left
tff(fact_3977_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_3978_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_3979_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3980_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_3981_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_3982_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_3983_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_3984_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_3985_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Ya) )
    <=> ( ( Xa = zero_zero_nat )
        & ( Ya = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_3986_add__eq__0__iff__both__eq__0,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Ya) = zero_zero_nat )
    <=> ( ( Xa = zero_zero_nat )
        & ( Ya = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_3987_add__cancel__right__right,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_3988_add__cancel__right__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_3989_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_3990_add__cancel__right__right,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_3991_add__cancel__right__left,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_3992_add__cancel__right__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_3993_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_3994_add__cancel__right__left,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_3995_add__cancel__left__right,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = A )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_3996_add__cancel__left__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = A )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_3997_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = A )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_3998_add__cancel__left__right,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = A )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_3999_add__cancel__left__left,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = A )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_4000_add__cancel__left__left,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = A )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_4001_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = A )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_4002_add__cancel__left__left,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = A )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_4003_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_4004_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_4005_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_4006_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_4007_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_4008_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_4009_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_4010_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_4011_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_cancel_left
tff(fact_4012_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_cancel_left
tff(fact_4013_add__less__cancel__left,axiom,
    ! [C2: nat,A: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% add_less_cancel_left
tff(fact_4014_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_cancel_left
tff(fact_4015_add__less__cancel__right,axiom,
    ! [A: 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_cancel_right
tff(fact_4016_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_cancel_right
tff(fact_4017_add__less__cancel__right,axiom,
    ! [A: 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,A),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,A),B2) ) ).

% add_less_cancel_right
tff(fact_4018_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_cancel_right
tff(fact_4019_add__diff__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_4020_add__diff__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_4021_add__diff__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_4022_diff__add__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_4023_diff__add__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_4024_diff__add__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_4025_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) ).

% add_diff_cancel_left
tff(fact_4026_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) ).

% add_diff_cancel_left
tff(fact_4027_add__diff__cancel__left,axiom,
    ! [C2: nat,A: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ).

% add_diff_cancel_left
tff(fact_4028_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ).

% add_diff_cancel_left
tff(fact_4029_add__diff__cancel__left_H,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_4030_add__diff__cancel__left_H,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_4031_add__diff__cancel__left_H,axiom,
    ! [A: 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,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_4032_add__diff__cancel__left_H,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_4033_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) ).

% add_diff_cancel_right
tff(fact_4034_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) ).

% add_diff_cancel_right
tff(fact_4035_add__diff__cancel__right,axiom,
    ! [A: 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,A),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,A),B2) ).

% add_diff_cancel_right
tff(fact_4036_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ).

% add_diff_cancel_right
tff(fact_4037_add__diff__cancel__right_H,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_4038_add__diff__cancel__right_H,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_4039_add__diff__cancel__right_H,axiom,
    ! [A: 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,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_4040_add__diff__cancel__right_H,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_4041_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_4042_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_4043_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_4044_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_4045_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_4046_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_4047_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_4048_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_4049_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_4050_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_4051_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_4052_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_4053_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_4054_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_4055_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_4056_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_4057_of__nat__add,axiom,
    ! [M2: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_add
tff(fact_4058_of__nat__add,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_add
tff(fact_4059_of__nat__add,axiom,
    ! [M2: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_add
tff(fact_4060_of__nat__add,axiom,
    ! [M2: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_add
tff(fact_4061_abs__add__abs,axiom,
    ! [A: code_integer,B2: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2)) ).

% abs_add_abs
tff(fact_4062_abs__add__abs,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_add_abs
tff(fact_4063_abs__add__abs,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_add_abs
tff(fact_4064_abs__add__abs,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ).

% abs_add_abs
tff(fact_4065_abs__of__nat,axiom,
    ! [Na: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% abs_of_nat
tff(fact_4066_abs__of__nat,axiom,
    ! [Na: nat] : abs_abs_Code_integer(semiri4939895301339042750nteger(Na)) = semiri4939895301339042750nteger(Na) ).

% abs_of_nat
tff(fact_4067_abs__of__nat,axiom,
    ! [Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% abs_of_nat
tff(fact_4068_abs__of__nat,axiom,
    ! [Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% abs_of_nat
tff(fact_4069_add__le__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_4070_add__le__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_4071_add__le__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_4072_add__le__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_4073_add__le__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_4074_add__le__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_4075_add__le__same__cancel2,axiom,
    ! [A: 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,A),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_4076_add__le__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_4077_le__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel1
tff(fact_4078_le__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel1
tff(fact_4079_le__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel1
tff(fact_4080_le__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel1
tff(fact_4081_le__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel2
tff(fact_4082_le__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel2
tff(fact_4083_le__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel2
tff(fact_4084_le__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel2
tff(fact_4085_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_4086_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_4087_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_4088_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_4089_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_4090_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_4091_add__less__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_4092_add__less__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_4093_add__less__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_4094_add__less__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_4095_add__less__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_4096_add__less__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_4097_add__less__same__cancel2,axiom,
    ! [A: 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,A),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_4098_add__less__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_4099_less__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel1
tff(fact_4100_less__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel1
tff(fact_4101_less__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel1
tff(fact_4102_less__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel1
tff(fact_4103_less__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel2
tff(fact_4104_less__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel2
tff(fact_4105_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel2
tff(fact_4106_less__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel2
tff(fact_4107_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_4108_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_4109_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_4110_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_4111_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_4112_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_4113_le__add__diff__inverse,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4114_le__add__diff__inverse,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4115_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4116_le__add__diff__inverse,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4117_le__add__diff__inverse2,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4118_le__add__diff__inverse2,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4119_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4120_le__add__diff__inverse2,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4121_diff__add__zero,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_4122_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_4123_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4124_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4125_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4126_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_4127_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_4128_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_4129_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_4130_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_4131_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_4132_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_4133_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_4134_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_4135_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_4136_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_4137_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_4138_divide__le__0__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B2))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4139_divide__le__0__abs__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B2))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4140_zero__le__divide__abs__iff,axiom,
    ! [A: 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,A),aa_real_real(abs_abs_real,B2)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4141_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B2)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4142_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M2)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M2)) ).

% of_nat_Suc
tff(fact_4143_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2)) ).

% of_nat_Suc
tff(fact_4144_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M2)) ).

% of_nat_Suc
tff(fact_4145_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M2)) ).

% of_nat_Suc
tff(fact_4146_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M2)) ).

% of_nat_Suc
tff(fact_4147_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_4148_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_4149_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_4150_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),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,W2),Z) ) ) ).

% zless_nat_conj
tff(fact_4151_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% zle_add1_eq_le
tff(fact_4152_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_4153_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_4154_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_4155_abs__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B2))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))) ).

% abs_triangle_ineq
tff(fact_4156_abs__triangle__ineq,axiom,
    ! [A: 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,A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq
tff(fact_4157_abs__triangle__ineq,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))) ).

% abs_triangle_ineq
tff(fact_4158_abs__triangle__ineq,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq
tff(fact_4159_is__num__normalize_I1_J,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% is_num_normalize(1)
tff(fact_4160_is__num__normalize_I1_J,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ).

% is_num_normalize(1)
tff(fact_4161_is__num__normalize_I1_J,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% is_num_normalize(1)
tff(fact_4162_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_4163_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_4164_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: 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,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_4165_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_4166_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4167_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4168_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4169_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_4170_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_4171_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_4172_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_4173_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_4174_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,B2: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_4175_group__cancel_Oadd2,axiom,
    ! [B: rat,K: rat,B2: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_4176_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A: 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,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_4177_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_4178_add_Oassoc,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% add.assoc
tff(fact_4179_add_Oassoc,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ).

% add.assoc
tff(fact_4180_add_Oassoc,axiom,
    ! [A: 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,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% add.assoc
tff(fact_4181_add_Oassoc,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% add.assoc
tff(fact_4182_add_Oleft__cancel,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_4183_add_Oleft__cancel,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_4184_add_Oleft__cancel,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_4185_add_Oright__cancel,axiom,
    ! [B2: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_4186_add_Oright__cancel,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_4187_add_Oright__cancel,axiom,
    ! [B2: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_4188_add_Ocommute,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) ).

% add.commute
tff(fact_4189_add_Ocommute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) ).

% add.commute
tff(fact_4190_add_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) ).

% add.commute
tff(fact_4191_add_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) ).

% add.commute
tff(fact_4192_add_Oleft__commute,axiom,
    ! [B2: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% add.left_commute
tff(fact_4193_add_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ).

% add.left_commute
tff(fact_4194_add_Oleft__commute,axiom,
    ! [B2: nat,A: 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,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% add.left_commute
tff(fact_4195_add_Oleft__commute,axiom,
    ! [B2: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% add.left_commute
tff(fact_4196_add__left__imp__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_4197_add__left__imp__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_4198_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_4199_add__left__imp__eq,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_4200_add__right__imp__eq,axiom,
    ! [B2: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_4201_add__right__imp__eq,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_4202_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_4203_add__right__imp__eq,axiom,
    ! [B2: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_4204_abs__diff__le__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_4205_abs__diff__le__iff,axiom,
    ! [Xa: real,A: real,R2: 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,Xa),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_4206_abs__diff__le__iff,axiom,
    ! [Xa: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R2)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_4207_abs__diff__le__iff,axiom,
    ! [Xa: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_4208_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer,C2: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B2),plus_p5714425477246183910nteger(C2,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C2)),abs_abs_Code_integer(minus_8373710615458151222nteger(B2,D)))) ).

% abs_diff_triangle_ineq
tff(fact_4209_abs__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C2: real,D: 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,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_4210_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_4211_abs__diff__triangle__ineq,axiom,
    ! [A: int,B2: int,C2: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_4212_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))) ).

% abs_triangle_ineq4
tff(fact_4213_abs__triangle__ineq4,axiom,
    ! [A: 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,A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq4
tff(fact_4214_abs__triangle__ineq4,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))) ).

% abs_triangle_ineq4
tff(fact_4215_abs__triangle__ineq4,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq4
tff(fact_4216_abs__diff__less__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_4217_abs__diff__less__iff,axiom,
    ! [Xa: real,A: real,R2: 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,Xa),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_4218_abs__diff__less__iff,axiom,
    ! [Xa: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R2)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_4219_abs__diff__less__iff,axiom,
    ! [Xa: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_4220_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_4221_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_4222_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_4223_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_4224_abs__le__D1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% abs_le_D1
tff(fact_4225_abs__le__D1,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2) ) ).

% abs_le_D1
tff(fact_4226_abs__le__D1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% abs_le_D1
tff(fact_4227_abs__le__D1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% abs_le_D1
tff(fact_4228_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_4229_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_4230_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_4231_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_4232_abs__minus__commute,axiom,
    ! [A: code_integer,B2: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B2,A)) ).

% abs_minus_commute
tff(fact_4233_abs__minus__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)) ).

% abs_minus_commute
tff(fact_4234_abs__minus__commute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A)) ).

% abs_minus_commute
tff(fact_4235_abs__minus__commute,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)) ).

% abs_minus_commute
tff(fact_4236_add__le__imp__le__right,axiom,
    ! [A: 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_imp_le_right
tff(fact_4237_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_imp_le_right
tff(fact_4238_add__le__imp__le__right,axiom,
    ! [A: 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,A),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,A),B2) ) ).

% add_le_imp_le_right
tff(fact_4239_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_imp_le_right
tff(fact_4240_add__le__imp__le__left,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_imp_le_left
tff(fact_4241_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_imp_le_left
tff(fact_4242_add__le__imp__le__left,axiom,
    ! [C2: nat,A: 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),A)),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,A),B2) ) ).

% add_le_imp_le_left
tff(fact_4243_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_imp_le_left
tff(fact_4244_le__iff__add,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ? [C5: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C5) ) ).

% le_iff_add
tff(fact_4245_add__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ) ).

% add_right_mono
tff(fact_4246_add__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ) ).

% add_right_mono
tff(fact_4247_add__right__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ) ).

% add_right_mono
tff(fact_4248_add__right__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ) ).

% add_right_mono
tff(fact_4249_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ~ ! [C4: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% less_eqE
tff(fact_4250_add__left__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% add_left_mono
tff(fact_4251_add__left__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2)) ) ).

% add_left_mono
tff(fact_4252_add__left__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) ) ).

% add_left_mono
tff(fact_4253_add__left__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)) ) ).

% add_left_mono
tff(fact_4254_add__mono,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_mono
tff(fact_4255_add__mono,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_mono
tff(fact_4256_add__mono,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_mono
tff(fact_4257_add__mono,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_mono
tff(fact_4258_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4259_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4260_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4261_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_4262_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4263_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4264_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4265_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_4266_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4267_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4268_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4269_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_4270_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_4271_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_4272_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_4273_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_4274_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_4275_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_4276_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_4277_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_4278_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_4279_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_4280_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_4281_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_4282_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_4283_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_4284_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_4285_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4286_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4287_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4288_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_4289_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4290_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4291_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4292_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_4293_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4294_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4295_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4296_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_4297_add__strict__mono,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_strict_mono
tff(fact_4298_add__strict__mono,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_strict_mono
tff(fact_4299_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_strict_mono
tff(fact_4300_add__strict__mono,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_strict_mono
tff(fact_4301_add__strict__left__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% add_strict_left_mono
tff(fact_4302_add__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2)) ) ).

% add_strict_left_mono
tff(fact_4303_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) ) ).

% add_strict_left_mono
tff(fact_4304_add__strict__left__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)) ) ).

% add_strict_left_mono
tff(fact_4305_add__strict__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ) ).

% add_strict_right_mono
tff(fact_4306_add__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ) ).

% add_strict_right_mono
tff(fact_4307_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ) ).

% add_strict_right_mono
tff(fact_4308_add__strict__right__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ) ).

% add_strict_right_mono
tff(fact_4309_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_imp_less_left
tff(fact_4310_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_imp_less_left
tff(fact_4311_add__less__imp__less__left,axiom,
    ! [C2: nat,A: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% add_less_imp_less_left
tff(fact_4312_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_imp_less_left
tff(fact_4313_add__less__imp__less__right,axiom,
    ! [A: 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_imp_less_right
tff(fact_4314_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_imp_less_right
tff(fact_4315_add__less__imp__less__right,axiom,
    ! [A: 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,A),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,A),B2) ) ).

% add_less_imp_less_right
tff(fact_4316_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_imp_less_right
tff(fact_4317_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M: int] :
        ? [N: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(abs_abs_int,N))
          & aa_set_int_o(member_int(N),S) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_4318_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M: int] :
        ? [N: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),aa_int_int(abs_abs_int,N))
          & aa_set_int_o(member_int(N),S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_4319_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_4320_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_4321_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_4322_diff__eq__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = C2 )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2) ) ) ).

% diff_eq_eq
tff(fact_4323_diff__eq__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = C2 )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2) ) ) ).

% diff_eq_eq
tff(fact_4324_diff__eq__eq,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2) ) ) ).

% diff_eq_eq
tff(fact_4325_eq__diff__eq,axiom,
    ! [A: real,C2: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_4326_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_4327_eq__diff__eq,axiom,
    ! [A: int,C2: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_4328_add__diff__eq,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),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,A),B2)),C2) ).

% add_diff_eq
tff(fact_4329_add__diff__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C2) ).

% add_diff_eq
tff(fact_4330_add__diff__eq,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C2) ).

% add_diff_eq
tff(fact_4331_diff__diff__eq2,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),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,A),C2)),B2) ).

% diff_diff_eq2
tff(fact_4332_diff__diff__eq2,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B2) ).

% diff_diff_eq2
tff(fact_4333_diff__diff__eq2,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B2) ).

% diff_diff_eq2
tff(fact_4334_diff__add__eq,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B2) ).

% diff_add_eq
tff(fact_4335_diff__add__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B2) ).

% diff_add_eq
tff(fact_4336_diff__add__eq,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B2) ).

% diff_add_eq
tff(fact_4337_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),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,A),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_4338_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_4339_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_4340_add__implies__diff,axiom,
    ! [C2: real,B2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2) = A )
     => ( C2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) ) ) ).

% add_implies_diff
tff(fact_4341_add__implies__diff,axiom,
    ! [C2: rat,B2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2) = A )
     => ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) ) ) ).

% add_implies_diff
tff(fact_4342_add__implies__diff,axiom,
    ! [C2: nat,B2: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ) ) ).

% add_implies_diff
tff(fact_4343_add__implies__diff,axiom,
    ! [C2: int,B2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ) ) ).

% add_implies_diff
tff(fact_4344_diff__diff__eq,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% diff_diff_eq
tff(fact_4345_diff__diff__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) ).

% diff_diff_eq
tff(fact_4346_diff__diff__eq,axiom,
    ! [A: 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,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% diff_diff_eq
tff(fact_4347_diff__diff__eq,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% diff_diff_eq
tff(fact_4348_abs__add__one__gt__zero,axiom,
    ! [Xa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xa))) ).

% abs_add_one_gt_zero
tff(fact_4349_abs__add__one__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xa))) ).

% abs_add_one_gt_zero
tff(fact_4350_abs__add__one__gt__zero,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,Xa))) ).

% abs_add_one_gt_zero
tff(fact_4351_abs__add__one__gt__zero,axiom,
    ! [Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xa))) ).

% abs_add_one_gt_zero
tff(fact_4352_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_4353_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_4354_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_4355_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_4356_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_4357_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_4358_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_4359_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_4360_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_4361_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_4362_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_4363_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_4364_nat__abs__int__diff,axiom,
    ! [A: nat,B2: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) ).

% nat_abs_int_diff
tff(fact_4365_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2))) ).

% abs_triangle_ineq2
tff(fact_4366_abs__triangle__ineq2,axiom,
    ! [A: 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,A)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% abs_triangle_ineq2
tff(fact_4367_abs__triangle__ineq2,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2))) ).

% abs_triangle_ineq2
tff(fact_4368_abs__triangle__ineq2,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))) ).

% abs_triangle_ineq2
tff(fact_4369_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2))) ).

% abs_triangle_ineq3
tff(fact_4370_abs__triangle__ineq3,axiom,
    ! [A: 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,A)),aa_real_real(abs_abs_real,B2)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% abs_triangle_ineq3
tff(fact_4371_abs__triangle__ineq3,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2))) ).

% abs_triangle_ineq3
tff(fact_4372_abs__triangle__ineq3,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))) ).

% abs_triangle_ineq3
tff(fact_4373_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))),abs_abs_Code_integer(minus_8373710615458151222nteger(B2,A))) ).

% abs_triangle_ineq2_sym
tff(fact_4374_abs__triangle__ineq2__sym,axiom,
    ! [A: 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,A)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))) ).

% abs_triangle_ineq2_sym
tff(fact_4375_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A))) ).

% abs_triangle_ineq2_sym
tff(fact_4376_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A))) ).

% abs_triangle_ineq2_sym
tff(fact_4377_nonzero__abs__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_4378_nonzero__abs__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_4379_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_4380_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_4381_nat__mono,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Ya)) ) ).

% nat_mono
tff(fact_4382_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_4383_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => aa_nat_o(P,aa_int_nat(nat2,X)) ) ) ).

% all_nat
tff(fact_4384_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
          & aa_nat_o(P,aa_int_nat(nat2,X)) ) ) ).

% ex_nat
tff(fact_4385_add__decreasing,axiom,
    ! [A: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B2) ) ) ).

% add_decreasing
tff(fact_4386_add__decreasing,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B2) ) ) ).

% add_decreasing
tff(fact_4387_add__decreasing,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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,A),C2)),B2) ) ) ).

% add_decreasing
tff(fact_4388_add__decreasing,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B2) ) ) ).

% add_decreasing
tff(fact_4389_add__increasing,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_increasing
tff(fact_4390_add__increasing,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% add_increasing
tff(fact_4391_add__increasing,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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,A),C2)) ) ) ).

% add_increasing
tff(fact_4392_add__increasing,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing
tff(fact_4393_add__decreasing2,axiom,
    ! [C2: real,A: 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,A),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_4394_add__decreasing2,axiom,
    ! [C2: rat,A: 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,A),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_4395_add__decreasing2,axiom,
    ! [C2: nat,A: 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,A),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_4396_add__decreasing2,axiom,
    ! [C2: int,A: 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,A),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_4397_add__increasing2,axiom,
    ! [C2: real,B2: real,A: 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),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_increasing2
tff(fact_4398_add__increasing2,axiom,
    ! [C2: rat,B2: rat,A: 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),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% add_increasing2
tff(fact_4399_add__increasing2,axiom,
    ! [C2: nat,B2: nat,A: 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),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing2
tff(fact_4400_add__increasing2,axiom,
    ! [C2: int,B2: int,A: 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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing2
tff(fact_4401_add__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,A),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_4402_add__nonneg__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_4403_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),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,A),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_4404_add__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_4405_add__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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,A),B2)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_4406_add__nonpos__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_4407_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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,A),B2)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_4408_add__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_4409_add__nonneg__eq__0__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Ya = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4410_add__nonneg__eq__0__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Ya = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4411_add__nonneg__eq__0__iff,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Ya) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Ya = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4412_add__nonneg__eq__0__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Ya) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Ya = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4413_add__nonpos__eq__0__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Ya = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4414_add__nonpos__eq__0__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Ya = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4415_add__nonpos__eq__0__iff,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Ya) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Ya = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4416_add__nonpos__eq__0__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Ya) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Ya = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4417_add__less__le__mono,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_4418_add__less__le__mono,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_4419_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_4420_add__less__le__mono,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_4421_add__le__less__mono,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_4422_add__le__less__mono,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_4423_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_4424_add__le__less__mono,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_4425_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4426_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4427_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4428_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_4429_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4430_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4431_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4432_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_4433_add__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_4434_add__neg__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_4435_add__neg__neg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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,A),B2)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_4436_add__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_4437_add__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% add_pos_pos
tff(fact_4438_add__pos__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_pos_pos
tff(fact_4439_add__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),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,A),B2)) ) ) ).

% add_pos_pos
tff(fact_4440_add__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_pos_pos
tff(fact_4441_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ~ ! [C4: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_4442_pos__add__strict,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% pos_add_strict
tff(fact_4443_pos__add__strict,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% pos_add_strict
tff(fact_4444_pos__add__strict,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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,A),C2)) ) ) ).

% pos_add_strict
tff(fact_4445_pos__add__strict,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% pos_add_strict
tff(fact_4446_add__less__zeroD,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_4447_add__less__zeroD,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_4448_add__less__zeroD,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Ya)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_4449_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A) = C2 )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4450_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_4451_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4452_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4453_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),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)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4454_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4455_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4456_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4457_le__add__diff,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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)),A)) ) ).

% le_add_diff
tff(fact_4458_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),A) = B2 ) ) ).

% diff_add
tff(fact_4459_le__diff__eq,axiom,
    ! [A: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),B2)),C2) ) ).

% le_diff_eq
tff(fact_4460_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C2) ) ).

% le_diff_eq
tff(fact_4461_le__diff__eq,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C2) ) ).

% le_diff_eq
tff(fact_4462_diff__le__eq,axiom,
    ! [A: 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,A),B2)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% diff_le_eq
tff(fact_4463_diff__le__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2)) ) ).

% diff_le_eq
tff(fact_4464_diff__le__eq,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)) ) ).

% diff_le_eq
tff(fact_4465_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4466_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4467_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4468_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4469_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Na: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4470_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Na: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4471_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Na: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4472_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Na: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4473_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_4474_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_4475_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_4476_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_4477_add__mono1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),one_one_real)) ) ).

% add_mono1
tff(fact_4478_add__mono1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),one_one_rat)) ) ).

% add_mono1
tff(fact_4479_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),one_one_nat)) ) ).

% add_mono1
tff(fact_4480_add__mono1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)) ) ).

% add_mono1
tff(fact_4481_diff__less__eq,axiom,
    ! [A: 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,A),B2)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% diff_less_eq
tff(fact_4482_diff__less__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B2)) ) ).

% diff_less_eq
tff(fact_4483_diff__less__eq,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)) ) ).

% diff_less_eq
tff(fact_4484_less__diff__eq,axiom,
    ! [A: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),B2)),C2) ) ).

% less_diff_eq
tff(fact_4485_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C2) ) ).

% less_diff_eq
tff(fact_4486_less__diff__eq,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C2) ) ).

% less_diff_eq
tff(fact_4487_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4488_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4489_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4490_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4491_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_4492_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_4493_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_4494_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z)
    <=> ? [N: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% zle_iff_zadd
tff(fact_4495_dbl__inc__def,axiom,
    ! [Xa: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xa) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Xa),Xa)),one_one_complex) ).

% dbl_inc_def
tff(fact_4496_dbl__inc__def,axiom,
    ! [Xa: real] : aa_real_real(neg_nu8295874005876285629c_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa)),one_one_real) ).

% dbl_inc_def
tff(fact_4497_dbl__inc__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xa) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa)),one_one_rat) ).

% dbl_inc_def
tff(fact_4498_dbl__inc__def,axiom,
    ! [Xa: int] : neg_nu5851722552734809277nc_int(Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa)),one_one_int) ).

% dbl_inc_def
tff(fact_4499_dense__eq0__I,axiom,
    ! [Xa: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),E) )
     => ( Xa = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_4500_dense__eq0__I,axiom,
    ! [Xa: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),E) )
     => ( Xa = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_4501_abs__div__pos,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,Xa)),Ya) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% abs_div_pos
tff(fact_4502_abs__div__pos,axiom,
    ! [Ya: real,Xa: 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,Xa)),Ya) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) ) ) ).

% abs_div_pos
tff(fact_4503_nat__mono__iff,axiom,
    ! [Z: int,W2: 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,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_mono_iff
tff(fact_4504_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_4505_add__neg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_4506_add__neg__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_4507_add__neg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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,A),B2)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_4508_add__neg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_4509_add__nonneg__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_4510_add__nonneg__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_4511_add__nonneg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),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,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_4512_add__nonneg__pos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_4513_add__nonpos__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_4514_add__nonpos__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_4515_add__nonpos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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,A),B2)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_4516_add__nonpos__neg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_4517_add__pos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_4518_add__pos__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_4519_add__pos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),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,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_4520_add__pos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_4521_add__strict__increasing,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_4522_add__strict__increasing,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_4523_add__strict__increasing,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_4524_add__strict__increasing,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_4525_add__strict__increasing2,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_4526_add__strict__increasing2,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_4527_add__strict__increasing2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_4528_add__strict__increasing2,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_4529_field__le__epsilon,axiom,
    ! [Xa: real,Ya: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ya),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% field_le_epsilon
tff(fact_4530_field__le__epsilon,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ya),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ).

% field_le_epsilon
tff(fact_4531_nat__le__iff,axiom,
    ! [Xa: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_4532_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_4533_int__eq__iff,axiom,
    ! [M2: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = Z )
    <=> ( ( M2 = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_4534_discrete,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B2) ) ).

% discrete
tff(fact_4535_discrete,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B2) ) ).

% discrete
tff(fact_4536_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_4537_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_4538_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_4539_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_4540_div__add__self2,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int) ) ) ).

% div_add_self2
tff(fact_4541_div__add__self2,axiom,
    ! [B2: nat,A: 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,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_4542_div__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int) ) ) ).

% div_add_self1
tff(fact_4543_div__add__self1,axiom,
    ! [B2: nat,A: 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),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_4544_gt__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B2) ) ).

% gt_half_sum
tff(fact_4545_gt__half__sum,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B2) ) ).

% gt_half_sum
tff(fact_4546_less__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_4547_less__half__sum,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_4548_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
    <=> ? [N: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% zless_iff_Suc_zadd
tff(fact_4549_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_4550_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% add1_zle_eq
tff(fact_4551_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_4552_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_4553_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_4554_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | 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,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_4555_nat__eq__iff2,axiom,
    ! [M2: nat,W2: int] :
      ( ( M2 = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M2),M2 = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_4556_nat__eq__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( aa_int_nat(nat2,W2) = M2 )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M2),M2 = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_4557_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N) )
           => aa_nat_o(P,N) )
        & ( 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_4558_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_4559_nat__diff__distrib_H,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Ya)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4560_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z4),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_4561_nat__div__distrib_H,axiom,
    ! [Ya: int,Xa: 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,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Ya)) ) ) ).

% nat_div_distrib'
tff(fact_4562_nat__div__distrib,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Ya)) ) ) ).

% nat_div_distrib
tff(fact_4563_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_4564_nat__less__iff,axiom,
    ! [W2: int,M2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),M2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M2)) ) ) ).

% nat_less_iff
tff(fact_4565_power__diff__power__eq,axiom,
    ! [A: int,M2: nat,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))) ) ) ).

% power_diff_power_eq
tff(fact_4566_power__diff__power__eq,axiom,
    ! [A: nat,M2: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))) ) ) ).

% power_diff_power_eq
tff(fact_4567_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4568_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4569_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4570_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4571_power__decreasing__iff,axiom,
    ! [B2: real,M2: nat,Na: 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),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_4572_power__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,Na: 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),M2)),aa_nat_rat(power_power_rat(B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_4573_power__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,Na: 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),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_4574_power__decreasing__iff,axiom,
    ! [B2: int,M2: nat,Na: 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),M2)),aa_nat_int(power_power_int(B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_4575_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(abs_abs_Code_integer(A),Na))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4576_zero__less__power__abs__iff,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na))
    <=> ( ( A != zero_zero_real )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4577_zero__less__power__abs__iff,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na))
    <=> ( ( A != zero_zero_rat )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4578_zero__less__power__abs__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na))
    <=> ( ( A != zero_zero_int )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4579_power__mono__iff,axiom,
    ! [A: real,B2: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_4580_power__mono__iff,axiom,
    ! [A: rat,B2: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B2),Na))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_4581_power__mono__iff,axiom,
    ! [A: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_4582_power__mono__iff,axiom,
    ! [A: int,B2: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_4583_power__increasing__iff,axiom,
    ! [B2: real,Xa: 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),Xa)),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,Xa),Ya) ) ) ).

% power_increasing_iff
tff(fact_4584_power__increasing__iff,axiom,
    ! [B2: rat,Xa: 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),Xa)),aa_nat_rat(power_power_rat(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya) ) ) ).

% power_increasing_iff
tff(fact_4585_power__increasing__iff,axiom,
    ! [B2: nat,Xa: 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),Xa)),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,Xa),Ya) ) ) ).

% power_increasing_iff
tff(fact_4586_power__increasing__iff,axiom,
    ! [B2: int,Xa: 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),Xa)),aa_nat_int(power_power_int(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya) ) ) ).

% power_increasing_iff
tff(fact_4587_power__strict__decreasing__iff,axiom,
    ! [B2: real,M2: nat,Na: 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),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4588_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,Na: 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),M2)),aa_nat_rat(power_power_rat(B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4589_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,Na: 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),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4590_power__strict__decreasing__iff,axiom,
    ! [B2: int,M2: nat,Na: 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),M2)),aa_nat_int(power_power_int(B2),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4591_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4592_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W2)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4593_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4594_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_4595_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4596_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4597_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4598_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_4599_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W2)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4600_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4601_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4602_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_4603_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N2: nat] : Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_4604_add__Suc__right,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) ).

% add_Suc_right
tff(fact_4605_add__is__0,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_4606_Nat_Oadd__0__right,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),zero_zero_nat) = M2 ).

% Nat.add_0_right
tff(fact_4607_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% nat_add_left_cancel_less
tff(fact_4608_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% nat_add_left_cancel_le
tff(fact_4609_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_4610_power__inject__exp,axiom,
    ! [A: real,M2: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) )
      <=> ( M2 = Na ) ) ) ).

% power_inject_exp
tff(fact_4611_power__inject__exp,axiom,
    ! [A: rat,M2: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),M2) = aa_nat_rat(power_power_rat(A),Na) )
      <=> ( M2 = Na ) ) ) ).

% power_inject_exp
tff(fact_4612_power__inject__exp,axiom,
    ! [A: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) )
      <=> ( M2 = Na ) ) ) ).

% power_inject_exp
tff(fact_4613_power__inject__exp,axiom,
    ! [A: int,M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),M2) = aa_nat_int(power_power_int(A),Na) )
      <=> ( M2 = Na ) ) ) ).

% power_inject_exp
tff(fact_4614_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Na)) = zero_zero_rat ).

% power_0_Suc
tff(fact_4615_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Na)) = zero_zero_int ).

% power_0_Suc
tff(fact_4616_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Na)) = zero_zero_nat ).

% power_0_Suc
tff(fact_4617_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Na)) = zero_zero_real ).

% power_0_Suc
tff(fact_4618_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Na)) = zero_zero_complex ).

% power_0_Suc
tff(fact_4619_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_4620_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_4621_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_4622_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_4623_add__gr__0,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% add_gr_0
tff(fact_4624_power__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_4625_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),M2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = zero_zero_nat )
        | ( Xa = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_4626_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_4627_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4628_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_4629_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_4630_power__strict__increasing__iff,axiom,
    ! [B2: real,Xa: 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),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_4631_power__strict__increasing__iff,axiom,
    ! [B2: rat,Xa: 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),Xa)),aa_nat_rat(power_power_rat(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_4632_power__strict__increasing__iff,axiom,
    ! [B2: nat,Xa: 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),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_4633_power__strict__increasing__iff,axiom,
    ! [B2: int,Xa: 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),Xa)),aa_nat_int(power_power_int(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_4634_power__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_4635_power__eq__0__iff,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_nat_int(power_power_int(A),Na) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_4636_power__eq__0__iff,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_4637_power__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_4638_power__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Na) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_4639_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4640_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4641_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4642_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4643_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4644_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B2: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_4645_add__Suc__shift,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M2)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(suc,Na)) ).

% add_Suc_shift
tff(fact_4646_add__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M2)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) ).

% add_Suc
tff(fact_4647_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_4648_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ).

% plus_nat.add_0
tff(fact_4649_add__eq__self__zero,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na) = M2 )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_4650_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_4651_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_4652_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_4653_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_4654_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_4655_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M2)) ) ).

% trans_less_add1
tff(fact_4656_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J)) ) ).

% trans_less_add2
tff(fact_4657_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% less_add_eq_less
tff(fact_4658_add__leE,axiom,
    ! [M2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_4659_le__add1,axiom,
    ! [Na: nat,M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2)) ).

% le_add1
tff(fact_4660_le__add2,axiom,
    ! [Na: nat,M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) ).

% le_add2
tff(fact_4661_add__leD1,axiom,
    ! [M2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% add_leD1
tff(fact_4662_add__leD2,axiom,
    ! [M2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_4663_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_4664_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_4665_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_4666_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M2)) ) ).

% trans_le_add1
tff(fact_4667_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J)) ) ).

% trans_le_add2
tff(fact_4668_nat__le__iff__add,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3) ) ).

% nat_le_iff_add
tff(fact_4669_diff__add__inverse2,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)),Na) = M2 ).

% diff_add_inverse2
tff(fact_4670_diff__add__inverse,axiom,
    ! [Na: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2)),Na) = M2 ).

% diff_add_inverse
tff(fact_4671_diff__cancel2,axiom,
    ! [M2: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) ).

% diff_cancel2
tff(fact_4672_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na) ).

% Nat.diff_cancel
tff(fact_4673_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_4674_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),K)) ) ).

% power_gt_expt
tff(fact_4675_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_4676_add__is__1,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_4677_one__is__add,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na) )
    <=> ( ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_4678_real__arch__pow,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N2)) ) ).

% real_arch_pow
tff(fact_4679_less__natE,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ~ ! [Q3: nat] : Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q3)) ) ).

% less_natE
tff(fact_4680_less__add__Suc1,axiom,
    ! [I: nat,M2: 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),M2))) ).

% less_add_Suc1
tff(fact_4681_less__add__Suc2,axiom,
    ! [I: nat,M2: 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,M2),I))) ).

% less_add_Suc2
tff(fact_4682_less__iff__Suc__add,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3)) ) ).

% less_iff_Suc_add
tff(fact_4683_less__imp__Suc__add,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ? [K2: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)) ) ).

% less_imp_Suc_add
tff(fact_4684_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_4685_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M2: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M4)),aa_nat_nat(F,N2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M2)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K))) ) ).

% mono_nat_linear_lb
tff(fact_4686_diff__add__0,axiom,
    ! [Na: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2)) = zero_zero_nat ).

% diff_add_0
tff(fact_4687_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_4688_add__diff__inverse__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_4689_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ).

% Suc_eq_plus1_left
tff(fact_4690_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_4691_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ).

% Suc_eq_plus1
tff(fact_4692_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_4693_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_4694_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4695_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4696_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4697_real__arch__pow__inv,axiom,
    ! [Ya: real,Xa: 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,Xa),one_one_real)
       => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N2)),Ya) ) ) ).

% real_arch_pow_inv
tff(fact_4698_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_4699_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_4700_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_4701_card__Un__le,axiom,
    ! [A2: set_complex,B: set_complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(sup_sup_set_complex(A2,B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A2)),finite_card_complex(B))) ).

% card_Un_le
tff(fact_4702_card__Un__le,axiom,
    ! [A2: set_set_nat,B: set_set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(sup_sup_set_set_nat(A2,B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A2)),finite_card_set_nat(B))) ).

% card_Un_le
tff(fact_4703_card__Un__le,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(sup_su793286257634532545t_unit(A2,B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B))) ).

% card_Un_le
tff(fact_4704_card__Un__le,axiom,
    ! [A2: set_list_nat,B: set_list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(sup_sup_set_list_nat(A2,B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A2)),finite_card_list_nat(B))) ).

% card_Un_le
tff(fact_4705_card__Un__le,axiom,
    ! [A2: set_nat,B: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(A2)),finite_card_nat(B))) ).

% card_Un_le
tff(fact_4706_card__Un__le,axiom,
    ! [A2: set_o,B: set_o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_o(A2)),finite_card_o(B))) ).

% card_Un_le
tff(fact_4707_card__Un__le,axiom,
    ! [A2: set_int,B: set_int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A2)),finite_card_int(B))) ).

% card_Un_le
tff(fact_4708_add__eq__if,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na) = $ite(M2 = zero_zero_nat,Na,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),Na))) ).

% add_eq_if
tff(fact_4709_nat__less__real__le,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M2)) ) ).

% nat_less_real_le
tff(fact_4710_nat__le__real__less,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M2)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_4711_card__Un__Int,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A2)),finite_card_set_nat(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(sup_sup_set_set_nat(A2,B))),finite_card_set_nat(inf_inf_set_set_nat(A2,B))) ) ) ) ).

% card_Un_Int
tff(fact_4712_card__Un__Int,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( finite4290736615968046902t_unit(B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite410649719033368117t_unit(sup_su793286257634532545t_unit(A2,B))),finite410649719033368117t_unit(inf_in4660618365625256667t_unit(A2,B))) ) ) ) ).

% card_Un_Int
tff(fact_4713_card__Un__Int,axiom,
    ! [A2: set_list_nat,B: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A2)),finite_card_list_nat(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(sup_sup_set_list_nat(A2,B))),finite_card_list_nat(inf_inf_set_list_nat(A2,B))) ) ) ) ).

% card_Un_Int
tff(fact_4714_card__Un__Int,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A2)),finite_card_complex(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(sup_sup_set_complex(A2,B))),finite_card_complex(inf_inf_set_complex(A2,B))) ) ) ) ).

% card_Un_Int
tff(fact_4715_card__Un__Int,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite121521170596916366d_enat(sup_su4489774667511045786d_enat(A2,B))),finite121521170596916366d_enat(inf_in8357106775501769908d_enat(A2,B))) ) ) ) ).

% card_Un_Int
tff(fact_4716_card__Un__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(A2)
     => ( finite6177210948735845034at_nat(B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(A2)),finite711546835091564841at_nat(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B))),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B))) ) ) ) ).

% card_Un_Int
tff(fact_4717_card__Un__Int,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(A2)),finite_card_nat(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B))),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B))) ) ) ) ).

% card_Un_Int
tff(fact_4718_card__Un__Int,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_o(A2)),finite_card_o(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B))),finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B))) ) ) ) ).

% card_Un_Int
tff(fact_4719_card__Un__Int,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A2)),finite_card_int(B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B))) ) ) ) ).

% card_Un_Int
tff(fact_4720_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_4721_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_4722_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_4723_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_4724_card__Un__disjoint,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( finite4290736615968046902t_unit(B)
       => ( ( inf_in4660618365625256667t_unit(A2,B) = bot_bo3957492148770167129t_unit )
         => ( finite410649719033368117t_unit(sup_su793286257634532545t_unit(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite410649719033368117t_unit(A2)),finite410649719033368117t_unit(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4725_card__Un__disjoint,axiom,
    ! [A2: set_list_nat,B: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(B)
       => ( ( inf_inf_set_list_nat(A2,B) = bot_bot_set_list_nat )
         => ( finite_card_list_nat(sup_sup_set_list_nat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A2)),finite_card_list_nat(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4726_card__Un__disjoint,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B)
       => ( ( inf_inf_set_complex(A2,B) = bot_bot_set_complex )
         => ( finite_card_complex(sup_sup_set_complex(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A2)),finite_card_complex(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4727_card__Un__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(A2)
     => ( finite6177210948735845034at_nat(B)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat )
         => ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(A2)),finite711546835091564841at_nat(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4728_card__Un__disjoint,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat )
         => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(A2)),finite_card_nat(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4729_card__Un__disjoint,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int )
         => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A2)),finite_card_int(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4730_card__Un__disjoint,axiom,
    ! [A2: set_o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B)
       => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o )
         => ( finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_o(A2)),finite_card_o(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4731_card__Un__disjoint,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B)
       => ( ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat )
         => ( finite_card_set_nat(sup_sup_set_set_nat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A2)),finite_card_set_nat(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4732_card__Un__disjoint,axiom,
    ! [A2: set_real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(finite_finite_real,B)
       => ( ( inf_inf_set_real(A2,B) = bot_bot_set_real )
         => ( finite_card_real(sup_sup_set_real(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_real(A2)),finite_card_real(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4733_card__Un__disjoint,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B)
       => ( ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat )
         => ( finite121521170596916366d_enat(sup_su4489774667511045786d_enat(A2,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4734_power__not__zero,axiom,
    ! [A: rat,Na: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Na) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_4735_power__not__zero,axiom,
    ! [A: int,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Na) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_4736_power__not__zero,axiom,
    ! [A: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_4737_power__not__zero,axiom,
    ! [A: real,Na: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_4738_power__not__zero,axiom,
    ! [A: complex,Na: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Na) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_4739_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_4740_zero__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_power
tff(fact_4741_zero__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_power
tff(fact_4742_zero__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_le_power
tff(fact_4743_zero__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_power
tff(fact_4744_power__mono,axiom,
    ! [A: real,B2: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na)) ) ) ).

% power_mono
tff(fact_4745_power__mono,axiom,
    ! [A: rat,B2: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B2),Na)) ) ) ).

% power_mono
tff(fact_4746_power__mono,axiom,
    ! [A: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na)) ) ) ).

% power_mono
tff(fact_4747_power__mono,axiom,
    ! [A: int,B2: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na)) ) ) ).

% power_mono
tff(fact_4748_zero__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_less_power
tff(fact_4749_zero__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_less_power
tff(fact_4750_zero__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_less_power
tff(fact_4751_zero__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_less_power
tff(fact_4752_one__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% one_le_power
tff(fact_4753_one__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% one_le_power
tff(fact_4754_one__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% one_le_power
tff(fact_4755_one__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ).

% one_le_power
tff(fact_4756_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_4757_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_4758_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_4759_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_4760_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_4761_power__less__imp__less__base,axiom,
    ! [A: real,Na: 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,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na))
     => ( 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,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_4762_power__less__imp__less__base,axiom,
    ! [A: rat,Na: nat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B2),Na))
     => ( 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,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_4763_power__less__imp__less__base,axiom,
    ! [A: nat,Na: 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,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na))
     => ( 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,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_4764_power__less__imp__less__base,axiom,
    ! [A: int,Na: nat,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na))
     => ( 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,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_4765_power__le__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),one_one_real) ) ) ).

% power_le_one
tff(fact_4766_power__le__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),one_one_rat) ) ) ).

% power_le_one
tff(fact_4767_power__le__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat) ) ) ).

% power_le_one
tff(fact_4768_power__le__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),one_one_int) ) ) ).

% power_le_one
tff(fact_4769_power__le__imp__le__base,axiom,
    ! [A: real,Na: 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,A),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,Na)))
     => ( 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,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_4770_power__le__imp__le__base,axiom,
    ! [A: rat,Na: nat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,Na)))
     => ( 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,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_4771_power__le__imp__le__base,axiom,
    ! [A: nat,Na: 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,A),aa_nat_nat(suc,Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,Na)))
     => ( 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,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_4772_power__le__imp__le__base,axiom,
    ! [A: int,Na: nat,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(B2),aa_nat_nat(suc,Na)))
     => ( 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,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_4773_power__inject__base,axiom,
    ! [A: real,Na: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,Na)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_4774_power__inject__base,axiom,
    ! [A: rat,Na: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,Na)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_4775_power__inject__base,axiom,
    ! [A: nat,Na: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_4776_power__inject__base,axiom,
    ! [A: int,Na: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_nat_int(power_power_int(B2),aa_nat_nat(suc,Na)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_4777_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_4778_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_4779_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_4780_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_4781_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_4782_power__gt1,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4783_power__gt1,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4784_power__gt1,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4785_power__gt1,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4786_power__less__imp__less__exp,axiom,
    ! [A: real,M2: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4787_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4788_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4789_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4790_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_4791_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_4792_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_4793_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_4794_zero__le__power__abs,axiom,
    ! [A: code_integer,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(abs_abs_Code_integer(A),Na)) ).

% zero_le_power_abs
tff(fact_4795_zero__le__power__abs,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) ).

% zero_le_power_abs
tff(fact_4796_zero__le__power__abs,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na)) ).

% zero_le_power_abs
tff(fact_4797_zero__le__power__abs,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na)) ).

% zero_le_power_abs
tff(fact_4798_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).

% power_increasing
tff(fact_4799_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_increasing
tff(fact_4800_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_4801_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_increasing
tff(fact_4802_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = zero_zero_rat ) ) ).

% zero_power
tff(fact_4803_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(power_power_int(zero_zero_int),Na) = zero_zero_int ) ) ).

% zero_power
tff(fact_4804_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = zero_zero_nat ) ) ).

% zero_power
tff(fact_4805_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = zero_zero_real ) ) ).

% zero_power
tff(fact_4806_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = zero_zero_complex ) ) ).

% zero_power
tff(fact_4807_power__Suc__le__self,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4808_power__Suc__le__self,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4809_power__Suc__le__self,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4810_power__Suc__le__self,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4811_power__Suc__less__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_4812_power__Suc__less__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_4813_power__Suc__less__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_4814_power__Suc__less__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_4815_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4816_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4817_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4818_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4819_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4820_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4821_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4822_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4823_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4824_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4825_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4826_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4827_power__eq__imp__eq__base,axiom,
    ! [A: real,Na: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4828_power__eq__imp__eq__base,axiom,
    ! [A: rat,Na: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B2),Na) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4829_power__eq__imp__eq__base,axiom,
    ! [A: nat,Na: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4830_power__eq__imp__eq__base,axiom,
    ! [A: int,Na: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B2),Na) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4831_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4832_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: rat,B2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B2),Na) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4833_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4834_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B2),Na) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4835_self__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% self_le_power
tff(fact_4836_self__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% self_le_power
tff(fact_4837_self__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% self_le_power
tff(fact_4838_self__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Na)) ) ) ).

% self_le_power
tff(fact_4839_one__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% one_less_power
tff(fact_4840_one__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% one_less_power
tff(fact_4841_one__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% one_less_power
tff(fact_4842_one__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ) ).

% one_less_power
tff(fact_4843_power__diff,axiom,
    ! [A: rat,Na: nat,M2: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_diff
tff(fact_4844_power__diff,axiom,
    ! [A: int,Na: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_diff
tff(fact_4845_power__diff,axiom,
    ! [A: nat,Na: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_diff
tff(fact_4846_power__diff,axiom,
    ! [A: real,Na: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_diff
tff(fact_4847_power__diff,axiom,
    ! [A: complex,Na: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),Na)) ) ) ) ).

% power_diff
tff(fact_4848_power__strict__mono,axiom,
    ! [A: real,B2: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4849_power__strict__mono,axiom,
    ! [A: rat,B2: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B2),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4850_power__strict__mono,axiom,
    ! [A: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4851_power__strict__mono,axiom,
    ! [A: int,B2: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4852_lemma__interval,axiom,
    ! [A: real,Xa: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B2)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [Y2: 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,Xa),Y2))),D5)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_4853_realpow__pos__nth,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Na) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_4854_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),Na) = A )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Na) = A ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_4855_lemma__interval__lt,axiom,
    ! [A: real,Xa: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B2)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [Y2: 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,Xa),Y2))),D5)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_4856_realpow__pos__nth2,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Na)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_4857_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_4858_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_4859_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_4860_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M5)
     => ? [N2: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X4,M5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X4)),N2) ) ) ).

% finite_maxlen
tff(fact_4861_finite__maxlen,axiom,
    ! [M5: set_list_o] :
      ( finite_finite_list_o(M5)
     => ? [N2: nat] :
        ! [X4: list_o] :
          ( member_list_o(X4,M5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X4)),N2) ) ) ).

% finite_maxlen
tff(fact_4862_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( finite8100373058378681591st_nat(M5)
     => ? [N2: nat] :
        ! [X4: list_nat] :
          ( member_list_nat(X4,M5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X4)),N2) ) ) ).

% finite_maxlen
tff(fact_4863_sin__bound__lemma,axiom,
    ! [Xa: real,Ya: real,U: real,V: real] :
      ( ( Xa = 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,Xa),U)),Ya))),V) ) ) ).

% sin_bound_lemma
tff(fact_4864_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B2: nat] :
      ( ! [A3: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B4)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B4),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B2) ) ) ) ).

% Euclid_induct
tff(fact_4865_add__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_4866_add__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_4867_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_4868_add__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_4869_ln__root,axiom,
    ! [Na: nat,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Na),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_4870_log__of__power__le,axiom,
    ! [M2: nat,B2: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na))
     => ( 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),M2)
         => 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,M2))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_4871_decr__lemma,axiom,
    ! [D: int,Xa: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_4872_incr__lemma,axiom,
    ! [D: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_4873_linear__plus__1__le__power,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)),Na)) ) ).

% linear_plus_1_le_power
tff(fact_4874_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_4875_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_4876_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_4877_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_4878_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_4879_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_4880_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_4881_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_4882_mult__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_4883_mult__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_4884_mult__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_4885_mult__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_4886_mult__cancel__left,axiom,
    ! [C2: real,A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_4887_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_4888_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_4889_mult__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_4890_mult__cancel__right,axiom,
    ! [A: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_4891_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_4892_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_4893_mult__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_4894_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_4895_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_4896_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_4897_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_4898_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_4899_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_4900_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_4901_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_4902_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_4903_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_4904_of__nat__mult,axiom,
    ! [M2: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_mult
tff(fact_4905_of__nat__mult,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mult
tff(fact_4906_of__nat__mult,axiom,
    ! [M2: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mult
tff(fact_4907_of__nat__mult,axiom,
    ! [M2: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_mult
tff(fact_4908_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_4909_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_4910_mult__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_4911_mult__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_4912_mult__cancel__left2,axiom,
    ! [C2: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_4913_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_4914_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_4915_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_4916_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_4917_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_4918_mult__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_4919_mult__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_4920_mult__cancel__right2,axiom,
    ! [A: complex,C2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_4921_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_4922_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_4923_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_4924_sum__squares__eq__zero__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)) = zero_zero_real )
    <=> ( ( Xa = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4925_sum__squares__eq__zero__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Ya)) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4926_sum__squares__eq__zero__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Ya)) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4927_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) = $ite(C2 = zero_zero_rat,zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_4928_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_4929_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B2)) = $ite(C2 = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_4930_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4931_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4932_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4933_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4934_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4935_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(B2),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4936_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4937_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A: 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4938_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B2),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4939_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4940_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A: 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4941_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(C2),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4942_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4943_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4944_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4945_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4946_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4947_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4948_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4949_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A: 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,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4950_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4951_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4952_div__mult__mult1,axiom,
    ! [C2: int,A: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ).

% div_mult_mult1
tff(fact_4953_div__mult__mult1,axiom,
    ! [C2: nat,A: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ).

% div_mult_mult1
tff(fact_4954_div__mult__mult2,axiom,
    ! [C2: int,A: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ).

% div_mult_mult2
tff(fact_4955_div__mult__mult2,axiom,
    ! [C2: nat,A: 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,A),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,A),B2) ) ) ).

% div_mult_mult2
tff(fact_4956_div__mult__mult1__if,axiom,
    ! [C2: int,A: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_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,A),B2)) ).

% div_mult_mult1_if
tff(fact_4957_div__mult__mult1__if,axiom,
    ! [C2: nat,A: 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),A)),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,A),B2)) ).

% div_mult_mult1_if
tff(fact_4958_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa))
    <=> ( Xa = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_4959_real__root__Suc__0,axiom,
    ! [Xa: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xa) = Xa ).

% real_root_Suc_0
tff(fact_4960_root__0,axiom,
    ! [Xa: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xa) = zero_zero_real ).

% root_0
tff(fact_4961_real__root__eq__iff,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = aa_real_real(aa_nat_fun_real_real(root,Na),Ya) )
      <=> ( Xa = Ya ) ) ) ).

% real_root_eq_iff
tff(fact_4962_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4963_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4964_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4965_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4966_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4967_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4968_div__mult__self1,axiom,
    ! [B2: int,A: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self1
tff(fact_4969_div__mult__self1,axiom,
    ! [B2: nat,A: 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,A),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,A),B2)) ) ) ).

% div_mult_self1
tff(fact_4970_div__mult__self2,axiom,
    ! [B2: int,A: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self2
tff(fact_4971_div__mult__self2,axiom,
    ! [B2: nat,A: 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,A),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,A),B2)) ) ) ).

% div_mult_self2
tff(fact_4972_div__mult__self3,axiom,
    ! [B2: int,C2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self3
tff(fact_4973_div__mult__self3,axiom,
    ! [B2: nat,C2: nat,A: 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)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self3
tff(fact_4974_div__mult__self4,axiom,
    ! [B2: int,C2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self4
tff(fact_4975_div__mult__self4,axiom,
    ! [B2: nat,C2: nat,A: 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)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self4
tff(fact_4976_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_4977_real__root__less__iff,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ).

% real_root_less_iff
tff(fact_4978_real__root__le__iff,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ).

% real_root_le_iff
tff(fact_4979_real__root__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_4980_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = one_one_real )
      <=> ( Xa = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_4981_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_4982_log__less__cancel__iff,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Ya))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_4983_log__less__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_4984_one__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_4985_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_4986_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_4987_real__root__gt__0__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya) ) ) ).

% real_root_gt_0_iff
tff(fact_4988_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_4989_real__root__le__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_4990_real__root__ge__0__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya) ) ) ).

% real_root_ge_0_iff
tff(fact_4991_real__root__gt__1__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ya) ) ) ).

% real_root_gt_1_iff
tff(fact_4992_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_4993_real__root__le__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_4994_real__root__ge__1__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Ya) ) ) ).

% real_root_ge_1_iff
tff(fact_4995_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_4996_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_4997_one__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_4998_log__le__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_4999_log__le__cancel__iff,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Ya))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_5000_real__root__pow__pos2,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_5001_log__pow__cancel,axiom,
    ! [A: real,B2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B2)) = aa_nat_real(semiri5074537144036343181t_real,B2) ) ) ) ).

% log_pow_cancel
tff(fact_5002_mult_Oleft__commute,axiom,
    ! [B2: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% mult.left_commute
tff(fact_5003_mult_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ).

% mult.left_commute
tff(fact_5004_mult_Oleft__commute,axiom,
    ! [B2: nat,A: 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,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% mult.left_commute
tff(fact_5005_mult_Oleft__commute,axiom,
    ! [B2: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% mult.left_commute
tff(fact_5006_mult_Ocommute,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A) ).

% mult.commute
tff(fact_5007_mult_Ocommute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A) ).

% mult.commute
tff(fact_5008_mult_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A) ).

% mult.commute
tff(fact_5009_mult_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),A) ).

% mult.commute
tff(fact_5010_mult_Oassoc,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% mult.assoc
tff(fact_5011_mult_Oassoc,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ).

% mult.assoc
tff(fact_5012_mult_Oassoc,axiom,
    ! [A: 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,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% mult.assoc
tff(fact_5013_mult_Oassoc,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% mult.assoc
tff(fact_5014_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_5015_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_5016_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: 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,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_5017_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_5018_mult__not__zero,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_5019_mult__not__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_5020_mult__not__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_5021_mult__not__zero,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_5022_divisors__zero,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_5023_divisors__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_5024_divisors__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_5025_divisors__zero,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_5026_no__zero__divisors,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_5027_no__zero__divisors,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_5028_no__zero__divisors,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_5029_no__zero__divisors,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_5030_mult__left__cancel,axiom,
    ! [C2: real,A: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_5031_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_5032_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_5033_mult__left__cancel,axiom,
    ! [C2: int,A: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_5034_mult__right__cancel,axiom,
    ! [C2: real,A: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_5035_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_5036_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_5037_mult__right__cancel,axiom,
    ! [C2: int,A: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_5038_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_5039_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_5040_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_5041_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_5042_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_5043_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_5044_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_5045_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_5046_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_5047_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_5048_mult__of__nat__commute,axiom,
    ! [Xa: nat,Ya: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),aa_nat_rat(semiri681578069525770553at_rat,Xa)) ).

% mult_of_nat_commute
tff(fact_5049_mult__of__nat__commute,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(semiri1316708129612266289at_nat,Xa)) ).

% mult_of_nat_commute
tff(fact_5050_mult__of__nat__commute,axiom,
    ! [Xa: nat,Ya: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),Ya) = aa_int_int(aa_int_fun_int_int(times_times_int,Ya),aa_nat_int(semiri1314217659103216013at_int,Xa)) ).

% mult_of_nat_commute
tff(fact_5051_mult__of__nat__commute,axiom,
    ! [Xa: nat,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),aa_nat_real(semiri5074537144036343181t_real,Xa)) ).

% mult_of_nat_commute
tff(fact_5052_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B2: real,C2: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B2 )
          & ( C2 != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C2)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5053_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B2: rat,C2: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B2 )
          & ( C2 != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5054_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B2: nat,C2: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B2 )
          & ( C2 != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5055_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B2: int,C2: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B2 )
          & ( C2 != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5056_log__base__root,axiom,
    ! [Na: nat,B2: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Na),B2)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) ) ) ) ).

% log_base_root
tff(fact_5057_log__mult,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Ya)) ) ) ) ) ) ).

% log_mult
tff(fact_5058_log__nat__power,axiom,
    ! [Xa: real,B2: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) ) ) ).

% log_nat_power
tff(fact_5059_real__root__pos__pos__le,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_5060_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5061_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5062_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5063_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5064_zero__le__mult__iff,axiom,
    ! [A: 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,A),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_5065_zero__le__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_5066_zero__le__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_5067_mult__nonneg__nonpos2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5068_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5069_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5070_mult__nonneg__nonpos2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5071_mult__nonpos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_5072_mult__nonpos__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_5073_mult__nonpos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_5074_mult__nonpos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_5075_mult__nonneg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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,A),B2)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_5076_mult__nonneg__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_5077_mult__nonneg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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,A),B2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_5078_mult__nonneg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_5079_mult__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,A),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5080_mult__nonneg__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5081_mult__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),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,A),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5082_mult__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5083_split__mult__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_5084_split__mult__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_5085_split__mult__neg__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_5086_split__mult__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_5087_mult__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_5088_mult__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_5089_mult__le__0__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_5090_mult__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_right_mono
tff(fact_5091_mult__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ) ) ).

% mult_right_mono
tff(fact_5092_mult__right__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ) ).

% mult_right_mono
tff(fact_5093_mult__right__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ) ) ).

% mult_right_mono
tff(fact_5094_mult__right__mono__neg,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_5095_mult__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_5096_mult__right__mono__neg,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( 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(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_5097_mult__left__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_left_mono
tff(fact_5098_mult__left__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) ) ) ).

% mult_left_mono
tff(fact_5099_mult__left__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% mult_left_mono
tff(fact_5100_mult__left__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ).

% mult_left_mono
tff(fact_5101_mult__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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,A),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_5102_mult__nonpos__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_5103_mult__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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(times_times_int,A),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_5104_mult__left__mono__neg,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_5105_mult__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_5106_mult__left__mono__neg,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( 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(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_5107_split__mult__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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,A),B2)) ) ).

% split_mult_pos_le
tff(fact_5108_split__mult__pos__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ).

% split_mult_pos_le
tff(fact_5109_split__mult__pos__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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(times_times_int,A),B2)) ) ).

% split_mult_pos_le
tff(fact_5110_zero__le__square,axiom,
    ! [A: 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,A),A)) ).

% zero_le_square
tff(fact_5111_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_5112_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_5113_mult__mono_H,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_5114_mult__mono_H,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_5115_mult__mono_H,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_5116_mult__mono_H,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_5117_mult__mono,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_5118_mult__mono,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( 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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_5119_mult__mono,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( 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,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_5120_mult__mono,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( 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(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_5121_mult__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,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,A),B2)) ) ) ).

% mult_neg_neg
tff(fact_5122_mult__neg__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_neg_neg
tff(fact_5123_mult__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_neg_neg
tff(fact_5124_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_5125_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_5126_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_5127_mult__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_5128_mult__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_5129_mult__less__0__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_5130_mult__neg__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_5131_mult__neg__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_5132_mult__neg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_5133_mult__neg__pos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_5134_mult__pos__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_5135_mult__pos__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_5136_mult__pos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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,A),B2)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_5137_mult__pos__neg,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_5138_mult__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_5139_mult__pos__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_5140_mult__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),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,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_5141_mult__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_5142_mult__pos__neg2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,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),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_5143_mult__pos__neg2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_5144_mult__pos__neg2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_5145_mult__pos__neg2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_5146_zero__less__mult__iff,axiom,
    ! [A: 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,A),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_5147_zero__less__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_5148_zero__less__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_5149_zero__less__mult__pos,axiom,
    ! [A: 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,A),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ).

% zero_less_mult_pos
tff(fact_5150_zero__less__mult__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ).

% zero_less_mult_pos
tff(fact_5151_zero__less__mult__pos,axiom,
    ! [A: 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,A),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ) ).

% zero_less_mult_pos
tff(fact_5152_zero__less__mult__pos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ).

% zero_less_mult_pos
tff(fact_5153_zero__less__mult__pos2,axiom,
    ! [B2: real,A: 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),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_5154_zero__less__mult__pos2,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_5155_zero__less__mult__pos2,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_5156_zero__less__mult__pos2,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_5157_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_5158_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: 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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_5159_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: 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(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_5160_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_5161_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: 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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_5162_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: 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(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_5163_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_5164_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_5165_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( 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(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_5166_mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_5167_mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_5168_mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_5169_mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_5170_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),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),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_5171_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_5172_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_5173_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_5174_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_5175_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( 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(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_5176_mult__strict__right__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_5177_mult__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_5178_mult__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_5179_mult__strict__right__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_5180_mult__less__cancel__right__disj,axiom,
    ! [A: 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,A),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,A),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),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_5181_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_5182_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_5183_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5184_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5185_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),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_5186_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5187_less__1__mult,axiom,
    ! [M2: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M2),Na)) ) ) ).

% less_1_mult
tff(fact_5188_less__1__mult,axiom,
    ! [M2: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M2),Na)) ) ) ).

% less_1_mult
tff(fact_5189_less__1__mult,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) ) ) ).

% less_1_mult
tff(fact_5190_less__1__mult,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M2),Na)) ) ) ).

% less_1_mult
tff(fact_5191_frac__eq__eq,axiom,
    ! [Ya: rat,Z: rat,Xa: rat,W2: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Ya) ) ) ) ) ).

% frac_eq_eq
tff(fact_5192_frac__eq__eq,axiom,
    ! [Ya: real,Z: real,Xa: real,W2: real] :
      ( ( Ya != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Ya) ) ) ) ) ).

% frac_eq_eq
tff(fact_5193_frac__eq__eq,axiom,
    ! [Ya: complex,Z: complex,Xa: complex,W2: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xa),Ya) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xa),Z) = aa_complex_complex(times_times_complex(W2),Ya) ) ) ) ) ).

% frac_eq_eq
tff(fact_5194_divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) = A )
    <=> $ite(C2 != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_5195_divide__eq__eq,axiom,
    ! [B2: real,C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A )
    <=> $ite(C2 != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_5196_divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = A )
    <=> $ite(C2 != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A),C2),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_5197_eq__divide__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = B2,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_5198_eq__divide__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( A = 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,A),C2) = B2,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_5199_eq__divide__eq,axiom,
    ! [A: complex,B2: complex,C2: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(A),C2) = B2,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_5200_divide__eq__imp,axiom,
    ! [C2: rat,B2: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_5201_divide__eq__imp,axiom,
    ! [C2: real,B2: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_5202_divide__eq__imp,axiom,
    ! [C2: complex,B2: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B2 = aa_complex_complex(times_times_complex(A),C2) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_5203_eq__divide__imp,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = B2 )
       => ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_5204_eq__divide__imp,axiom,
    ! [C2: real,A: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B2 )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_5205_eq__divide__imp,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = B2 )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_5206_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B2: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) = A )
      <=> ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_5207_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B2: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A )
      <=> ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_5208_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B2: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = A )
      <=> ( B2 = aa_complex_complex(times_times_complex(A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_5209_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_5210_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_5211_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
      <=> ( aa_complex_complex(times_times_complex(A),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_5212_abs__mult__less,axiom,
    ! [A: code_integer,C2: code_integer,B2: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B2)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))),times_3573771949741848930nteger(C2,D)) ) ) ).

% abs_mult_less
tff(fact_5213_abs__mult__less,axiom,
    ! [A: real,C2: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B2)),D)
       => 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,A)),aa_real_real(abs_abs_real,B2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),D)) ) ) ).

% abs_mult_less
tff(fact_5214_abs__mult__less,axiom,
    ! [A: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B2)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),D)) ) ) ).

% abs_mult_less
tff(fact_5215_abs__mult__less,axiom,
    ! [A: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B2)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D)) ) ) ).

% abs_mult_less
tff(fact_5216_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_5217_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B2: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
         => ( ( B2 != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa) = 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,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_5218_log__root,axiom,
    ! [Na: nat,A: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_nat_fun_real_real(root,Na),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),A)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_5219_real__root__less__mono,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Ya)) ) ) ).

% real_root_less_mono
tff(fact_5220_real__root__le__mono,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Ya)) ) ) ).

% real_root_le_mono
tff(fact_5221_real__root__power,axiom,
    ! [Na: nat,Xa: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),K) ) ) ).

% real_root_power
tff(fact_5222_real__root__abs,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(abs_abs_real,Xa)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_abs
tff(fact_5223_mult__less__le__imp__less,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5224_mult__less__le__imp__less,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5225_mult__less__le__imp__less,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5226_mult__less__le__imp__less,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5227_mult__le__less__imp__less,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5228_mult__le__less__imp__less,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5229_mult__le__less__imp__less,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5230_mult__le__less__imp__less,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5231_mult__right__le__imp__le,axiom,
    ! [A: 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,A),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,A),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_5232_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_5233_mult__right__le__imp__le,axiom,
    ! [A: 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,A),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,A),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_5234_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_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,A),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_5235_mult__left__le__imp__le,axiom,
    ! [C2: real,A: 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),A)),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,A),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_5236_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_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,A),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_5237_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: 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),A)),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,A),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_5238_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_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,A),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_5239_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_5240_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: 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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_5241_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: 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(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_5242_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: 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),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_5243_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: 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(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_5244_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: 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(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_5245_mult__less__cancel__right,axiom,
    ! [A: 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,A),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,A),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),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_5246_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_5247_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_5248_mult__strict__mono_H,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5249_mult__strict__mono_H,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5250_mult__strict__mono_H,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5251_mult__strict__mono_H,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5252_mult__right__less__imp__less,axiom,
    ! [A: 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,A),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,A),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_5253_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_5254_mult__right__less__imp__less,axiom,
    ! [A: 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,A),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,A),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_5255_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_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,A),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_5256_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),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),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_5257_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_5258_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_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,A),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),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_5259_mult__strict__mono,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( 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,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5260_mult__strict__mono,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( 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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5261_mult__strict__mono,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( 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,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5262_mult__strict__mono,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( 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(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5263_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_5264_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_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,A),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_5265_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: 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),A)),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,A),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_5266_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_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,A),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_5267_mult__le__cancel__right,axiom,
    ! [A: 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,A),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,A),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),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_5268_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_5269_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_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,A),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),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_5270_mult__le__cancel__left,axiom,
    ! [C2: real,A: 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),A)),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,A),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),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_5271_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_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,A),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),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_5272_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_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,A),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),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_5273_mult__left__le,axiom,
    ! [C2: real,A: 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),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_5274_mult__left__le,axiom,
    ! [C2: rat,A: 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),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_5275_mult__left__le,axiom,
    ! [C2: nat,A: 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),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_5276_mult__left__le,axiom,
    ! [C2: int,A: 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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_5277_mult__le__one,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,A),B2)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_5278_mult__le__one,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),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(aa_rat_fun_rat_rat(times_times_rat,A),B2)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_5279_mult__le__one,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),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,A),B2)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_5280_mult__le__one,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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(aa_int_fun_int_int(times_times_int,A),B2)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_5281_mult__right__le__one__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa),Ya)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_5282_mult__right__le__one__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),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(aa_rat_fun_rat_rat(times_times_rat,Xa),Ya)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_5283_mult__right__le__one__le,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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(aa_int_fun_int_int(times_times_int,Xa),Ya)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_5284_mult__left__le__one__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_5285_mult__left__le__one__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),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(aa_rat_fun_rat_rat(times_times_rat,Ya),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_5286_mult__left__le__one__le,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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(aa_int_fun_int_int(times_times_int,Ya),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_5287_sum__squares__le__zero__iff,axiom,
    ! [Xa: 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,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya))),zero_zero_real)
    <=> ( ( Xa = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_5288_sum__squares__le__zero__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Ya))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_5289_sum__squares__le__zero__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Ya))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_5290_sum__squares__ge__zero,axiom,
    ! [Xa: 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,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya))) ).

% sum_squares_ge_zero
tff(fact_5291_sum__squares__ge__zero,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Ya))) ).

% sum_squares_ge_zero
tff(fact_5292_sum__squares__ge__zero,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Ya))) ).

% sum_squares_ge_zero
tff(fact_5293_sum__squares__gt__zero__iff,axiom,
    ! [Xa: 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,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)))
    <=> ( ( Xa != zero_zero_real )
        | ( Ya != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_5294_sum__squares__gt__zero__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Ya)))
    <=> ( ( Xa != zero_zero_rat )
        | ( Ya != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_5295_sum__squares__gt__zero__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Ya)))
    <=> ( ( Xa != zero_zero_int )
        | ( Ya != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_5296_not__sum__squares__lt__zero,axiom,
    ! [Xa: 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,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_5297_not__sum__squares__lt__zero,axiom,
    ! [Xa: rat,Ya: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Ya))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_5298_not__sum__squares__lt__zero,axiom,
    ! [Xa: int,Ya: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Ya))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_5299_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: 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,A),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,A),B2)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_5300_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: 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,A),aa_int_int(aa_int_fun_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,A),B2)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_5301_divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),A)
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_5302_divide__less__eq,axiom,
    ! [B2: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A)
    <=> $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,A),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,A),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_5303_less__divide__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_5304_less__divide__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),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,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_5305_neg__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_5306_neg__divide__less__eq,axiom,
    ! [C2: real,B2: real,A: 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)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_5307_neg__less__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_5308_neg__less__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_5309_pos__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_5310_pos__divide__less__eq,axiom,
    ! [C2: real,B2: real,A: 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)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_5311_pos__less__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_5312_pos__less__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_5313_mult__imp__div__pos__less,axiom,
    ! [Ya: rat,Xa: 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,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Ya))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_5314_mult__imp__div__pos__less,axiom,
    ! [Ya: real,Xa: 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,Xa),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,Xa),Ya)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_5315_mult__imp__less__div__pos,axiom,
    ! [Ya: rat,Z: rat,Xa: 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(aa_rat_fun_rat_rat(times_times_rat,Z),Ya)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% mult_imp_less_div_pos
tff(fact_5316_mult__imp__less__div__pos,axiom,
    ! [Ya: real,Z: real,Xa: 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)),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) ) ) ).

% mult_imp_less_div_pos
tff(fact_5317_divide__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_5318_divide__strict__left__mono,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( 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,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_5319_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_5320_divide__strict__left__mono__neg,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_5321_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_5322_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_5323_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_5324_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E2)),D))
    <=> 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),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_5325_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_5326_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_5327_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B2) = $ite(Z = zero_zero_rat,B2,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_5328_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),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,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_5329_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B2: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B2) = $ite(Z = zero_zero_complex,B2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_5330_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B2: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_5331_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B2: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = $ite(Z = zero_zero_real,A,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,A),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_5332_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B2: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_5333_add__frac__eq,axiom,
    ! [Ya: rat,Z: rat,Xa: rat,W2: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Ya))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z)) ) ) ) ).

% add_frac_eq
tff(fact_5334_add__frac__eq,axiom,
    ! [Ya: real,Z: real,Xa: real,W2: 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,Xa),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z)) ) ) ) ).

% add_frac_eq
tff(fact_5335_add__frac__eq,axiom,
    ! [Ya: complex,Z: complex,Xa: complex,W2: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Ya)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xa),Z)),aa_complex_complex(times_times_complex(W2),Ya))),aa_complex_complex(times_times_complex(Ya),Z)) ) ) ) ).

% add_frac_eq
tff(fact_5336_add__frac__num,axiom,
    ! [Ya: rat,Xa: rat,Z: rat] :
      ( ( Ya != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Ya))),Ya) ) ) ).

% add_frac_num
tff(fact_5337_add__frac__num,axiom,
    ! [Ya: real,Xa: 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,Xa),Ya)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))),Ya) ) ) ).

% add_frac_num
tff(fact_5338_add__frac__num,axiom,
    ! [Ya: complex,Xa: complex,Z: complex] :
      ( ( Ya != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Ya)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(times_times_complex(Z),Ya))),Ya) ) ) ).

% add_frac_num
tff(fact_5339_add__num__frac,axiom,
    ! [Ya: rat,Z: rat,Xa: rat] :
      ( ( Ya != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Ya))),Ya) ) ) ).

% add_num_frac
tff(fact_5340_add__num__frac,axiom,
    ! [Ya: real,Z: real,Xa: real] :
      ( ( Ya != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))),Ya) ) ) ).

% add_num_frac
tff(fact_5341_add__num__frac,axiom,
    ! [Ya: complex,Z: complex,Xa: complex] :
      ( ( Ya != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xa),Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(times_times_complex(Z),Ya))),Ya) ) ) ).

% add_num_frac
tff(fact_5342_add__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ya),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),Ya)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_5343_add__divide__eq__iff,axiom,
    ! [Z: real,Xa: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa),Z)),Ya)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_5344_add__divide__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(divide1717551699836669952omplex(Ya),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xa),Z)),Ya)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_5345_divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_5346_divide__add__eq__iff,axiom,
    ! [Z: real,Xa: 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,Xa),Z)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_5347_divide__add__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Z)),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_5348_less__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E2)),D))
    <=> 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),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_5349_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_5350_less__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_5351_less__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E2)),D))
    <=> 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,A),B2)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_5352_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_5353_less__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_5354_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B2: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_5355_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B2: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = $ite(Z = zero_zero_real,A,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,A),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_5356_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B2: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_5357_diff__frac__eq,axiom,
    ! [Ya: rat,Z: rat,Xa: rat,W2: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Ya))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_5358_diff__frac__eq,axiom,
    ! [Ya: real,Z: real,Xa: real,W2: 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,Xa),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_5359_diff__frac__eq,axiom,
    ! [Ya: complex,Z: complex,Xa: complex,W2: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Ya)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Z)),aa_complex_complex(times_times_complex(W2),Ya))),aa_complex_complex(times_times_complex(Ya),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_5360_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Ya),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),Ya)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_5361_diff__divide__eq__iff,axiom,
    ! [Z: real,Xa: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),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,Xa),Z)),Ya)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_5362_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xa),aa_complex_complex(divide1717551699836669952omplex(Ya),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Z)),Ya)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_5363_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z)),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_5364_divide__diff__eq__iff,axiom,
    ! [Z: real,Xa: 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,Xa),Z)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_5365_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Z)),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xa),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_5366_ex__less__of__nat__mult,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_5367_ex__less__of__nat__mult,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [N2: 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,N2)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_5368_power__less__power__Suc,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_less_power_Suc
tff(fact_5369_power__less__power__Suc,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_less_power_Suc
tff(fact_5370_power__less__power__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_less_power_Suc
tff(fact_5371_power__less__power__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_less_power_Suc
tff(fact_5372_power__gt1__lemma,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_gt1_lemma
tff(fact_5373_power__gt1__lemma,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_gt1_lemma
tff(fact_5374_power__gt1__lemma,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_gt1_lemma
tff(fact_5375_power__gt1__lemma,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_gt1_lemma
tff(fact_5376_abs__eq__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(times_3573771949741848930nteger(A,B2)) = times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2)) ) ) ).

% abs_eq_mult
tff(fact_5377_abs__eq__mult,axiom,
    ! [A: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% abs_eq_mult
tff(fact_5378_abs__eq__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),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(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% abs_eq_mult
tff(fact_5379_abs__eq__mult,axiom,
    ! [A: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ) ) ).

% abs_eq_mult
tff(fact_5380_abs__mult__pos,axiom,
    ! [Xa: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( times_3573771949741848930nteger(abs_abs_Code_integer(Ya),Xa) = abs_abs_Code_integer(times_3573771949741848930nteger(Ya,Xa)) ) ) ).

% abs_mult_pos
tff(fact_5381_abs__mult__pos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Ya)),Xa) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Xa)) ) ) ).

% abs_mult_pos
tff(fact_5382_abs__mult__pos,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Ya)),Xa) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Xa)) ) ) ).

% abs_mult_pos
tff(fact_5383_abs__mult__pos,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Ya)),Xa) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Xa)) ) ) ).

% abs_mult_pos
tff(fact_5384_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ! [Y2: real] :
        ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),Xa)) ) ).

% reals_Archimedean3
tff(fact_5385_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M2)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),Na) = one_one_int )
      <=> ( ( M2 = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_5386_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P1,X2)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z3: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P1,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_5387_plusinfinity,axiom,
    ! [D: int,P2: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P2,X2)
          <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z3: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P2,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P2,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_5388_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: 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,A),aa_int_int(aa_int_fun_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,A),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_5389_real__root__gt__zero,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_5390_real__root__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_5391_log__base__change,axiom,
    ! [A: real,B2: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B2)) ) ) ) ).

% log_base_change
tff(fact_5392_root__abs__power,axiom,
    ! [Na: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Na))) = aa_real_real(abs_abs_real,Ya) ) ) ).

% root_abs_power
tff(fact_5393_less__log__of__power,axiom,
    ! [B2: real,Na: nat,M2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na)),M2)
     => ( 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,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),M2)) ) ) ).

% less_log_of_power
tff(fact_5394_log__of__power__eq,axiom,
    ! [M2: nat,B2: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M2)) ) ) ) ).

% log_of_power_eq
tff(fact_5395_field__le__mult__one__interval,axiom,
    ! [Xa: 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),Xa)),Ya) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% field_le_mult_one_interval
tff(fact_5396_field__le__mult__one__interval,axiom,
    ! [Xa: 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(aa_rat_fun_rat_rat(times_times_rat,Z2),Xa)),Ya) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ).

% field_le_mult_one_interval
tff(fact_5397_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),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,A),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),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_5398_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),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,A),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),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_5399_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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,A),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),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_5400_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_5401_mult__less__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_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_5402_mult__less__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_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_5403_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),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),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_5404_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),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,A),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),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_5405_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),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,A),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),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_5406_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_5407_mult__less__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_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_5408_mult__less__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_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_5409_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),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,A),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),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_5410_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),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,A),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),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_5411_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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,A),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),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_5412_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_5413_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(aa_rat_fun_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_5414_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(aa_int_fun_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_5415_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),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),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_5416_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),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,A),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),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_5417_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),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,A),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),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_5418_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_5419_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(aa_rat_fun_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_5420_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(aa_int_fun_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_5421_divide__left__mono__neg,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_5422_divide__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_5423_mult__imp__le__div__pos,axiom,
    ! [Ya: real,Z: real,Xa: 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)),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) ) ) ).

% mult_imp_le_div_pos
tff(fact_5424_mult__imp__le__div__pos,axiom,
    ! [Ya: rat,Z: rat,Xa: 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(aa_rat_fun_rat_rat(times_times_rat,Z),Ya)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)) ) ) ).

% mult_imp_le_div_pos
tff(fact_5425_mult__imp__div__pos__le,axiom,
    ! [Ya: real,Xa: 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,Xa),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,Xa),Ya)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_5426_mult__imp__div__pos__le,axiom,
    ! [Ya: rat,Xa: 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,Xa),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Ya))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_5427_pos__le__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_5428_pos__le__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_5429_pos__divide__le__eq,axiom,
    ! [C2: real,B2: real,A: 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)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_5430_pos__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_5431_neg__le__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_5432_neg__le__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_5433_neg__divide__le__eq,axiom,
    ! [C2: real,B2: real,A: 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)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_5434_neg__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_5435_divide__left__mono,axiom,
    ! [B2: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( 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,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_left_mono
tff(fact_5436_divide__left__mono,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B2)) ) ) ) ).

% divide_left_mono
tff(fact_5437_le__divide__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),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,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_5438_le__divide__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_5439_divide__le__eq,axiom,
    ! [B2: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A)
    <=> $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,A),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,A),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_5440_divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),A)
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_5441_convex__bound__le,axiom,
    ! [Xa: real,A: real,Ya: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Ya))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_5442_convex__bound__le,axiom,
    ! [Xa: rat,A: rat,Ya: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Ya))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_5443_convex__bound__le,axiom,
    ! [Xa: int,A: int,Ya: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Ya))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_5444_frac__le__eq,axiom,
    ! [Ya: real,Z: real,Xa: real,W2: 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,Xa),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_5445_frac__le__eq,axiom,
    ! [Ya: rat,Z: rat,Xa: rat,W2: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Ya))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_5446_frac__less__eq,axiom,
    ! [Ya: rat,Z: rat,Xa: rat,W2: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Ya))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_5447_frac__less__eq,axiom,
    ! [Ya: real,Z: real,Xa: real,W2: 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,Xa),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_5448_power__Suc__less,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% power_Suc_less
tff(fact_5449_power__Suc__less,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_Suc_less
tff(fact_5450_power__Suc__less,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% power_Suc_less
tff(fact_5451_power__Suc__less,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))),aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_Suc_less
tff(fact_5452_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_5453_q__pos__lemma,axiom,
    ! [B8: int,Q4: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B8),Q4)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B8)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B8)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_5454_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q5: int,R2: int,B8: int,Q4: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B8),Q4)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B8),Q4)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B8)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B8)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B8),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_5455_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q5: int,R2: int,B8: int,Q4: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B8),Q4)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B8),Q4)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),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),B8)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B8),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_5456_unique__quotient__lemma,axiom,
    ! [B2: int,Q4: int,R4: int,Q5: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q4)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q5) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_5457_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q4: int,R4: int,Q5: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q4)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R2)
         => ( 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,Q5),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_5458_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_5459_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5460_ln__mult,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Ya)) ) ) ) ).

% ln_mult
tff(fact_5461_real__root__pos__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_5462_real__root__strict__increasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,N4),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_5463_real__root__decreasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_5464_real__root__pow__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_5465_real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_5466_real__root__pos__unique,axiom,
    ! [Na: nat,Ya: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Na) = Xa )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = Ya ) ) ) ) ).

% real_root_pos_unique
tff(fact_5467_log__divide,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Ya)) ) ) ) ) ) ).

% log_divide
tff(fact_5468_le__log__of__power,axiom,
    ! [B2: real,Na: nat,M2: 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),Na)),M2)
     => ( 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,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),M2)) ) ) ).

% le_log_of_power
tff(fact_5469_log__base__pow,axiom,
    ! [A: real,Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_5470_convex__bound__lt,axiom,
    ! [Xa: real,A: real,Ya: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Ya))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_5471_convex__bound__lt,axiom,
    ! [Xa: rat,A: rat,Ya: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Ya))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_5472_convex__bound__lt,axiom,
    ! [Xa: int,A: int,Ya: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Ya))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_5473_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S3)
         => 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,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_5474_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V),U))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_5475_power__eq__if,axiom,
    ! [P3: complex,M2: nat] :
      aa_nat_complex(power_power_complex(P3),M2) = $ite(M2 = 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,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_5476_power__eq__if,axiom,
    ! [P3: real,M2: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M2) = $ite(M2 = 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,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_5477_power__eq__if,axiom,
    ! [P3: rat,M2: nat] :
      aa_nat_rat(power_power_rat(P3),M2) = $ite(M2 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_5478_power__eq__if,axiom,
    ! [P3: nat,M2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M2) = $ite(M2 = 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,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_5479_power__eq__if,axiom,
    ! [P3: int,M2: nat] :
      aa_nat_int(power_power_int(P3),M2) = $ite(M2 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_5480_power__minus__mult,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% power_minus_mult
tff(fact_5481_power__minus__mult,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) ) ) ).

% power_minus_mult
tff(fact_5482_power__minus__mult,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% power_minus_mult
tff(fact_5483_power__minus__mult,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) ) ) ).

% power_minus_mult
tff(fact_5484_power__minus__mult,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_minus_mult
tff(fact_5485_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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)),Xa)),C2) )
         => ( Xa = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_5486_split__zdiv,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,I4) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_5487_int__div__neg__eq,axiom,
    ! [A: int,B2: int,Q5: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = Q5 ) ) ) ) ).

% int_div_neg_eq
tff(fact_5488_int__div__pos__eq,axiom,
    ! [A: int,B2: int,Q5: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = Q5 ) ) ) ) ).

% int_div_pos_eq
tff(fact_5489_ln__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_realpow
tff(fact_5490_real__root__increasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,N4),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_5491_log__of__power__less,axiom,
    ! [M2: nat,B2: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na))
     => ( 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),M2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_5492_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xa: 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),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ).

% mult_le_cancel_iff2
tff(fact_5493_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xa: 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(aa_rat_fun_rat_rat(times_times_rat,Z),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Ya))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ) ).

% mult_le_cancel_iff2
tff(fact_5494_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xa: 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(aa_int_fun_int_int(times_times_int,Z),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Ya))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya) ) ) ).

% mult_le_cancel_iff2
tff(fact_5495_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xa: 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,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ).

% mult_le_cancel_iff1
tff(fact_5496_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xa: 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(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya) ) ) ).

% mult_le_cancel_iff1
tff(fact_5497_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xa: 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(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya) ) ) ).

% mult_le_cancel_iff1
tff(fact_5498_mult__less__iff1,axiom,
    ! [Z: real,Xa: 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,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ).

% mult_less_iff1
tff(fact_5499_mult__less__iff1,axiom,
    ! [Z: rat,Xa: 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(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya) ) ) ).

% mult_less_iff1
tff(fact_5500_mult__less__iff1,axiom,
    ! [Z: int,Xa: 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(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Ya),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Ya) ) ) ).

% mult_less_iff1
tff(fact_5501_arctan__add,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xa)),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,Xa),Ya)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Ya)))) ) ) ) ).

% arctan_add
tff(fact_5502_root__powr__inverse,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_5503_split__root,axiom,
    ! [P: fun_real_o,Na: nat,Xa: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ! [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)),Na)) = Xa )
             => aa_real_o(P,Y4) ) ) ) ) ).

% split_root
tff(fact_5504_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_5505_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_5506_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_5507_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = uminus_uminus_int(one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5508_add_Oinverse__inverse,axiom,
    ! [A: int] : uminus_uminus_int(uminus_uminus_int(A)) = A ).

% add.inverse_inverse
tff(fact_5509_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_5510_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_5511_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_5512_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_5513_neg__equal__iff__equal,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int(A) = uminus_uminus_int(B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_5514_neg__equal__iff__equal,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_5515_neg__equal__iff__equal,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_5516_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_5517_neg__equal__iff__equal,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_5518_neg__le__iff__le,axiom,
    ! [B2: real,A: 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,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% neg_le_iff_le
tff(fact_5519_neg__le__iff__le,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2) ) ).

% neg_le_iff_le
tff(fact_5520_neg__le__iff__le,axiom,
    ! [B2: rat,A: 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,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% neg_le_iff_le
tff(fact_5521_neg__le__iff__le,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(B2)),uminus_uminus_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% neg_le_iff_le
tff(fact_5522_compl__le__compl__iff,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xa)),uminus1532241313380277803et_int(Ya))
    <=> aa_set_int_o(ord_less_eq_set_int(Ya),Xa) ) ).

% compl_le_compl_iff
tff(fact_5523_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int(A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_5524_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_5525_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_5526_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_5527_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = uminus_uminus_int(A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_5528_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_5529_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_5530_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_5531_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int(A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_5532_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_5533_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_5534_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_5535_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_5536_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = uminus_uminus_int(A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5537_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5538_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5539_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5540_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5541_add_Oinverse__neutral,axiom,
    uminus_uminus_int(zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_5542_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_5543_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_5544_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_5545_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_5546_neg__less__iff__less,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(B2)),uminus_uminus_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% neg_less_iff_less
tff(fact_5547_neg__less__iff__less,axiom,
    ! [B2: real,A: 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,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% neg_less_iff_less
tff(fact_5548_neg__less__iff__less,axiom,
    ! [B2: rat,A: 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,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% neg_less_iff_less
tff(fact_5549_neg__less__iff__less,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2) ) ).

% neg_less_iff_less
tff(fact_5550_add__minus__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_5551_add__minus__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_5552_add__minus__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_5553_add__minus__cancel,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2)) = B2 ).

% add_minus_cancel
tff(fact_5554_add__minus__cancel,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_5555_minus__add__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = B2 ).

% minus_add_cancel
tff(fact_5556_minus__add__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = B2 ).

% minus_add_cancel
tff(fact_5557_minus__add__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = B2 ).

% minus_add_cancel
tff(fact_5558_minus__add__cancel,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B2)) = B2 ).

% minus_add_cancel
tff(fact_5559_minus__add__cancel,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B2)) = B2 ).

% minus_add_cancel
tff(fact_5560_minus__add__distrib,axiom,
    ! [A: int,B2: int] : uminus_uminus_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(A)),uminus_uminus_int(B2)) ).

% minus_add_distrib
tff(fact_5561_minus__add__distrib,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) ).

% minus_add_distrib
tff(fact_5562_minus__add__distrib,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_add_distrib
tff(fact_5563_minus__add__distrib,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% minus_add_distrib
tff(fact_5564_minus__add__distrib,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_add_distrib
tff(fact_5565_minus__diff__eq,axiom,
    ! [A: int,B2: int] : uminus_uminus_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A) ).

% minus_diff_eq
tff(fact_5566_minus__diff__eq,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A) ).

% minus_diff_eq
tff(fact_5567_minus__diff__eq,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A) ).

% minus_diff_eq
tff(fact_5568_minus__diff__eq,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B2)) = minus_8373710615458151222nteger(B2,A) ).

% minus_diff_eq
tff(fact_5569_minus__diff__eq,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B2)) = aa_complex_complex(minus_minus_complex(B2),A) ).

% minus_diff_eq
tff(fact_5570_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,uminus_uminus_int(A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_5571_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_5572_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_5573_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_5574_mult__cancel2,axiom,
    ! [M2: nat,K: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K) )
    <=> ( ( M2 = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_5575_mult__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( M2 = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_5576_mult__0__right,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_5577_mult__is__0,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_5578_sgn__0,axiom,
    sgn_sgn_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_5579_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_5580_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_5581_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_5582_sgn__0,axiom,
    sgn_sgn_int(zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_5583_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_5584_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_5585_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) )
    <=> ( ( M2 = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_5586_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) = one_one_nat )
    <=> ( ( M2 = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_5587_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_5588_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_5589_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_5590_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),uminus_uminus_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_5591_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_5592_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_5593_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_5594_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_5595_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_5596_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_5597_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_5598_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),uminus_uminus_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_5599_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_5600_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_5601_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_5602_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_5603_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),uminus_uminus_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_5604_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_5605_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_5606_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_5607_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_5608_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_5609_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_5610_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_5611_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),uminus_uminus_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_5612_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_5613_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_5614_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_5615_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_5616_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_5617_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_5618_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_5619_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),uminus_uminus_int(A)) = zero_zero_int ).

% add.right_inverse
tff(fact_5620_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_5621_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_5622_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_5623_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_5624_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_5625_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_5626_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_5627_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_5628_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_5629_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = uminus_uminus_int(A) ).

% diff_0
tff(fact_5630_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_5631_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_5632_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_5633_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_5634_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B2) = uminus_uminus_int(B2) ).

% verit_minus_simplify(3)
tff(fact_5635_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_5636_verit__minus__simplify_I3_J,axiom,
    ! [B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B2) = aa_rat_rat(uminus_uminus_rat,B2) ).

% verit_minus_simplify(3)
tff(fact_5637_verit__minus__simplify_I3_J,axiom,
    ! [B2: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ).

% verit_minus_simplify(3)
tff(fact_5638_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_5639_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,uminus_uminus_int(one_one_int)),Z) = uminus_uminus_int(Z) ).

% mult_minus1
tff(fact_5640_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_5641_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_5642_mult__minus1,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_5643_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_5644_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),uminus_uminus_int(one_one_int)) = uminus_uminus_int(Z) ).

% mult_minus1_right
tff(fact_5645_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_5646_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_5647_mult__minus1__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_5648_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_5649_diff__minus__eq__add,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),uminus_uminus_int(B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) ).

% diff_minus_eq_add
tff(fact_5650_diff__minus__eq__add,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) ).

% diff_minus_eq_add
tff(fact_5651_diff__minus__eq__add,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) ).

% diff_minus_eq_add
tff(fact_5652_diff__minus__eq__add,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = plus_p5714425477246183910nteger(A,B2) ).

% diff_minus_eq_add
tff(fact_5653_diff__minus__eq__add,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(plus_plus_complex(A),B2) ).

% diff_minus_eq_add
tff(fact_5654_uminus__add__conv__diff,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(A)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A) ).

% uminus_add_conv_diff
tff(fact_5655_uminus__add__conv__diff,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A) ).

% uminus_add_conv_diff
tff(fact_5656_uminus__add__conv__diff,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A) ).

% uminus_add_conv_diff
tff(fact_5657_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) = minus_8373710615458151222nteger(B2,A) ).

% uminus_add_conv_diff
tff(fact_5658_uminus__add__conv__diff,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(minus_minus_complex(B2),A) ).

% uminus_add_conv_diff
tff(fact_5659_inf__compl__bot__left1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left1
tff(fact_5660_inf__compl__bot__left1,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_5661_inf__compl__bot__left1,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(Xa)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_5662_inf__compl__bot__left1,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(Xa)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)) = bot_bot_set_o ).

% inf_compl_bot_left1
tff(fact_5663_inf__compl__bot__left1,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] : inf_inf_set_set_nat(uminus613421341184616069et_nat(Xa),inf_inf_set_set_nat(Xa,Ya)) = bot_bot_set_set_nat ).

% inf_compl_bot_left1
tff(fact_5664_inf__compl__bot__left1,axiom,
    ! [Xa: set_real,Ya: set_real] : inf_inf_set_real(uminus612125837232591019t_real(Xa),inf_inf_set_real(Xa,Ya)) = bot_bot_set_real ).

% inf_compl_bot_left1
tff(fact_5665_inf__compl__bot__left1,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] : inf_in8357106775501769908d_enat(uminus417252749190364093d_enat(Xa),inf_in8357106775501769908d_enat(Xa,Ya)) = bot_bo7653980558646680370d_enat ).

% inf_compl_bot_left1
tff(fact_5666_inf__compl__bot__left2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(Xa)),Ya)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left2
tff(fact_5667_inf__compl__bot__left2,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),Ya)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_5668_inf__compl__bot__left2,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(Xa)),Ya)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_5669_inf__compl__bot__left2,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(Xa)),Ya)) = bot_bot_set_o ).

% inf_compl_bot_left2
tff(fact_5670_inf__compl__bot__left2,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] : inf_inf_set_set_nat(Xa,inf_inf_set_set_nat(uminus613421341184616069et_nat(Xa),Ya)) = bot_bot_set_set_nat ).

% inf_compl_bot_left2
tff(fact_5671_inf__compl__bot__left2,axiom,
    ! [Xa: set_real,Ya: set_real] : inf_inf_set_real(Xa,inf_inf_set_real(uminus612125837232591019t_real(Xa),Ya)) = bot_bot_set_real ).

% inf_compl_bot_left2
tff(fact_5672_inf__compl__bot__left2,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] : inf_in8357106775501769908d_enat(Xa,inf_in8357106775501769908d_enat(uminus417252749190364093d_enat(Xa),Ya)) = bot_bo7653980558646680370d_enat ).

% inf_compl_bot_left2
tff(fact_5673_inf__compl__bot__right,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ya),uminus6524753893492686040at_nat(Xa))) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_right
tff(fact_5674_inf__compl__bot__right,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ya),uminus5710092332889474511et_nat(Xa))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_5675_inf__compl__bot__right,axiom,
    ! [Xa: set_int,Ya: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ya),uminus1532241313380277803et_int(Xa))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_5676_inf__compl__bot__right,axiom,
    ! [Xa: set_o,Ya: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ya),uminus_uminus_set_o(Xa))) = bot_bot_set_o ).

% inf_compl_bot_right
tff(fact_5677_inf__compl__bot__right,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] : inf_inf_set_set_nat(Xa,inf_inf_set_set_nat(Ya,uminus613421341184616069et_nat(Xa))) = bot_bot_set_set_nat ).

% inf_compl_bot_right
tff(fact_5678_inf__compl__bot__right,axiom,
    ! [Xa: set_real,Ya: set_real] : inf_inf_set_real(Xa,inf_inf_set_real(Ya,uminus612125837232591019t_real(Xa))) = bot_bot_set_real ).

% inf_compl_bot_right
tff(fact_5679_inf__compl__bot__right,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] : inf_in8357106775501769908d_enat(Xa,inf_in8357106775501769908d_enat(Ya,uminus417252749190364093d_enat(Xa))) = bot_bo7653980558646680370d_enat ).

% inf_compl_bot_right
tff(fact_5680_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(Xa)),Xa) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_5681_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),Xa) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_5682_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(Xa)),Xa) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_5683_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(Xa)),Xa) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_left
tff(fact_5684_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_set_nat] : inf_inf_set_set_nat(uminus613421341184616069et_nat(Xa),Xa) = bot_bot_set_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_5685_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(uminus612125837232591019t_real(Xa),Xa) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_left
tff(fact_5686_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_Extended_enat] : inf_in8357106775501769908d_enat(uminus417252749190364093d_enat(Xa),Xa) = bot_bo7653980558646680370d_enat ).

% boolean_algebra.conj_cancel_left
tff(fact_5687_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),uminus6524753893492686040at_nat(Xa)) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_5688_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),uminus5710092332889474511et_nat(Xa)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_5689_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),uminus1532241313380277803et_int(Xa)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_5690_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),uminus_uminus_set_o(Xa)) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_right
tff(fact_5691_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_set_nat] : inf_inf_set_set_nat(Xa,uminus613421341184616069et_nat(Xa)) = bot_bot_set_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_5692_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,uminus612125837232591019t_real(Xa)) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_right
tff(fact_5693_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_Extended_enat] : inf_in8357106775501769908d_enat(Xa,uminus417252749190364093d_enat(Xa)) = bot_bo7653980558646680370d_enat ).

% boolean_algebra.conj_cancel_right
tff(fact_5694_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,uminus_uminus_int(one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_5695_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_5696_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_5697_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_5698_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),sgn_sgn_Code_integer(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_5699_sgn__greater,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_greater
tff(fact_5700_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_5701_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),sgn_sgn_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_5702_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,sgn_sgn_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_5703_sgn__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_less
tff(fact_5704_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_less
tff(fact_5705_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,sgn_sgn_int(A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_5706_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(Xa)),uminus6524753893492686040at_nat(Ya)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_5707_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [Xa: set_nat,Ya: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Xa)),uminus5710092332889474511et_nat(Ya)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_5708_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [Xa: set_o,Ya: set_o] : uminus_uminus_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,uminus_uminus_set_o(Xa)),uminus_uminus_set_o(Ya)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_5709_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [Xa: set_int,Ya: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(Xa)),uminus1532241313380277803et_int(Ya)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_5710_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Xa),Ya)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(Xa)),uminus6524753893492686040at_nat(Ya)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_5711_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [Xa: set_nat,Ya: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Xa),Ya)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),uminus5710092332889474511et_nat(Ya)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_5712_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [Xa: set_o,Ya: set_o] : uminus_uminus_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Xa),Ya)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(Xa)),uminus_uminus_set_o(Ya)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_5713_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [Xa: set_int,Ya: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Xa),Ya)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(Xa)),uminus1532241313380277803et_int(Ya)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_5714_one__eq__mult__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_5715_mult__eq__1__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_5716_nat__0__less__mult__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_5717_mult__less__cancel2,axiom,
    ! [M2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% mult_less_cancel2
tff(fact_5718_powr__zero__eq__one,axiom,
    ! [Xa: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),zero_zero_real) = $ite(Xa = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_5719_mult__Suc__right,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) ).

% mult_Suc_right
tff(fact_5720_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_5721_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_5722_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_5723_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_5724_negative__eq__positive,axiom,
    ! [Na: nat,M2: nat] :
      ( ( uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,M2) )
    <=> ( ( Na = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_5725_powr__gt__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A))
    <=> ( Xa != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_5726_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_5727_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_5728_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_5729_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_5730_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_5731_powr__nonneg__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_5732_powr__less__cancel__iff,axiom,
    ! [Xa: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_5733_negative__zle,axiom,
    ! [Na: nat,M2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,M2)) ).

% negative_zle
tff(fact_5734_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_5735_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% zero_less_arctan_iff
tff(fact_5736_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_arctan_iff
tff(fact_5737_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_5738_dbl__inc__simps_I4_J,axiom,
    neg_nu5851722552734809277nc_int(uminus_uminus_int(one_one_int)) = uminus_uminus_int(one_one_int) ).

% dbl_inc_simps(4)
tff(fact_5739_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_5740_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_5741_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_5742_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_5743_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_5744_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_5745_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_5746_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_5747_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_5748_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),uminus_uminus_int(one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_5749_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_5750_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_5751_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_5752_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_5753_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(one_one_int)),uminus_uminus_int(one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_5754_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_5755_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_5756_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_5757_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_5758_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_5759_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_5760_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_5761_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = uminus_uminus_int(A) ) ) ).

% abs_of_nonpos
tff(fact_5762_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( sgn_sgn_Code_integer(A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_5763_sgn__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_5764_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_5765_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( sgn_sgn_int(A) = one_one_int ) ) ).

% sgn_pos
tff(fact_5766_one__le__mult__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ) ) ).

% one_le_mult_iff
tff(fact_5767_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_5768_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_5769_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_5770_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,sgn_sgn_int(A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_5771_mult__le__cancel2,axiom,
    ! [M2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% mult_le_cancel2
tff(fact_5772_div__mult__self1__is__m,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M2)),Na) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_5773_div__mult__self__is__m,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)),Na) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_5774_powr__eq__one__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa) = one_one_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_5775_powr__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),one_one_real) = Xa ) ) ).

% powr_one
tff(fact_5776_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),one_one_real) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_5777_negative__zless,axiom,
    ! [Na: nat,M2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),aa_nat_int(semiri1314217659103216013at_int,M2)) ).

% negative_zless
tff(fact_5778_powr__le__cancel__iff,axiom,
    ! [Xa: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_5779_nat__zminus__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_5780_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( sgn_sgn_int(A) = uminus_uminus_int(one_one_int) ) ) ).

% sgn_neg
tff(fact_5781_sgn__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_5782_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_5783_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( sgn_sgn_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_5784_powr__log__cancel,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_5785_log__powr__cancel,axiom,
    ! [A: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Ya)) = Ya ) ) ) ).

% log_powr_cancel
tff(fact_5786_sgn__not__eq__imp,axiom,
    ! [B2: int,A: int] :
      ( ( sgn_sgn_int(B2) != sgn_sgn_int(A) )
     => ( ( sgn_sgn_int(A) != zero_zero_int )
       => ( ( sgn_sgn_int(B2) != zero_zero_int )
         => ( sgn_sgn_int(A) = uminus_uminus_int(sgn_sgn_int(B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5787_sgn__not__eq__imp,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B2) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5788_sgn__not__eq__imp,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B2) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B2) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5789_sgn__not__eq__imp,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( sgn_sgn_Code_integer(B2) != sgn_sgn_Code_integer(A) )
     => ( ( sgn_sgn_Code_integer(A) != zero_z3403309356797280102nteger )
       => ( ( sgn_sgn_Code_integer(B2) != zero_z3403309356797280102nteger )
         => ( sgn_sgn_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,sgn_sgn_Code_integer(B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5790_equation__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = uminus_uminus_int(B2) )
    <=> ( B2 = uminus_uminus_int(A) ) ) ).

% equation_minus_iff
tff(fact_5791_equation__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B2) )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_5792_equation__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( B2 = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_5793_equation__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) )
    <=> ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_5794_equation__minus__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_5795_minus__equation__iff,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int(A) = B2 )
    <=> ( uminus_uminus_int(B2) = A ) ) ).

% minus_equation_iff
tff(fact_5796_minus__equation__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B2 )
    <=> ( aa_real_real(uminus_uminus_real,B2) = A ) ) ).

% minus_equation_iff
tff(fact_5797_minus__equation__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B2 )
    <=> ( aa_rat_rat(uminus_uminus_rat,B2) = A ) ) ).

% minus_equation_iff
tff(fact_5798_minus__equation__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B2 )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) = A ) ) ).

% minus_equation_iff
tff(fact_5799_minus__equation__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B2) = A ) ) ).

% minus_equation_iff
tff(fact_5800_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_5801_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_5802_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_5803_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_5804_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int(A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_5805_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_5806_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_5807_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_5808_sgn__0__0,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int(A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_5809_le__imp__neg__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A)) ) ).

% le_imp_neg_le
tff(fact_5810_le__imp__neg__le,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_5811_le__imp__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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,A)) ) ).

% le_imp_neg_le
tff(fact_5812_le__imp__neg__le,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(B2)),uminus_uminus_int(A)) ) ).

% le_imp_neg_le
tff(fact_5813_minus__le__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),A) ) ).

% minus_le_iff
tff(fact_5814_minus__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),A) ) ).

% minus_le_iff
tff(fact_5815_minus__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),A) ) ).

% minus_le_iff
tff(fact_5816_minus__le__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(B2)),A) ) ).

% minus_le_iff
tff(fact_5817_le__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A)) ) ).

% le_minus_iff
tff(fact_5818_le__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_5819_le__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),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,A)) ) ).

% le_minus_iff
tff(fact_5820_le__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),uminus_uminus_int(B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),uminus_uminus_int(A)) ) ).

% le_minus_iff
tff(fact_5821_compl__le__swap2,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Ya)),Xa)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xa)),Ya) ) ).

% compl_le_swap2
tff(fact_5822_compl__le__swap1,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),uminus1532241313380277803et_int(Xa))
     => aa_set_int_o(ord_less_eq_set_int(Xa),uminus1532241313380277803et_int(Ya)) ) ).

% compl_le_swap1
tff(fact_5823_compl__mono,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Ya)),uminus1532241313380277803et_int(Xa)) ) ).

% compl_mono
tff(fact_5824_less__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),uminus_uminus_int(B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),uminus_uminus_int(A)) ) ).

% less_minus_iff
tff(fact_5825_less__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A)) ) ).

% less_minus_iff
tff(fact_5826_less__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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,A)) ) ).

% less_minus_iff
tff(fact_5827_less__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_5828_minus__less__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(B2)),A) ) ).

% minus_less_iff
tff(fact_5829_minus__less__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),A) ) ).

% minus_less_iff
tff(fact_5830_minus__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),A) ) ).

% minus_less_iff
tff(fact_5831_minus__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),A) ) ).

% minus_less_iff
tff(fact_5832_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(B2)),uminus_uminus_int(A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5833_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5834_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),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,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5835_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5836_one__neq__neg__one,axiom,
    one_one_int != uminus_uminus_int(one_one_int) ).

% one_neq_neg_one
tff(fact_5837_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_5838_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_5839_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_5840_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_5841_is__num__normalize_I8_J,axiom,
    ! [A: int,B2: int] : uminus_uminus_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(B2)),uminus_uminus_int(A)) ).

% is_num_normalize(8)
tff(fact_5842_is__num__normalize_I8_J,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_5843_is__num__normalize_I8_J,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_5844_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_5845_is__num__normalize_I8_J,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_5846_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( uminus_uminus_int(A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(K)),uminus_uminus_int(A)) ) ) ).

% group_cancel.neg1
tff(fact_5847_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_5848_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_5849_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_5850_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_5851_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B2: int] : uminus_uminus_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(B2)),uminus_uminus_int(A)) ).

% add.inverse_distrib_swap
tff(fact_5852_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_5853_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_5854_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_5855_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_5856_minus__diff__commute,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(B2)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(A)),B2) ).

% minus_diff_commute
tff(fact_5857_minus__diff__commute,axiom,
    ! [B2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B2)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B2) ).

% minus_diff_commute
tff(fact_5858_minus__diff__commute,axiom,
    ! [B2: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ).

% minus_diff_commute
tff(fact_5859_minus__diff__commute,axiom,
    ! [B2: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) ).

% minus_diff_commute
tff(fact_5860_minus__diff__commute,axiom,
    ! [B2: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) ).

% minus_diff_commute
tff(fact_5861_Suc__mult__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na) )
    <=> ( M2 = Na ) ) ).

% Suc_mult_cancel1
tff(fact_5862_mult__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ).

% mult_0
tff(fact_5863_arctan__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% arctan_less_iff
tff(fact_5864_arctan__monotone,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Ya)) ) ).

% arctan_monotone
tff(fact_5865_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_5866_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_5867_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_5868_le__square,axiom,
    ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),M2)) ).

% le_square
tff(fact_5869_le__cube,axiom,
    ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),M2))) ).

% le_cube
tff(fact_5870_arctan__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% arctan_le_iff
tff(fact_5871_arctan__monotone_H,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Ya)) ) ).

% arctan_monotone'
tff(fact_5872_add__mult__distrib2,axiom,
    ! [K: nat,M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% add_mult_distrib2
tff(fact_5873_add__mult__distrib,axiom,
    ! [M2: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% add_mult_distrib
tff(fact_5874_diff__mult__distrib,axiom,
    ! [M2: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% diff_mult_distrib
tff(fact_5875_diff__mult__distrib2,axiom,
    ! [K: nat,M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% diff_mult_distrib2
tff(fact_5876_nat__mult__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ).

% nat_mult_1
tff(fact_5877_nat__mult__1__right,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ).

% nat_mult_1_right
tff(fact_5878_sgn__if,axiom,
    ! [Xa: int] :
      sgn_sgn_int(Xa) = $ite(
        Xa = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa),one_one_int,uminus_uminus_int(one_one_int)) ) ).

% sgn_if
tff(fact_5879_sgn__if,axiom,
    ! [Xa: real] :
      aa_real_real(sgn_sgn_real,Xa) = $ite(
        Xa = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_if
tff(fact_5880_sgn__if,axiom,
    ! [Xa: rat] :
      aa_rat_rat(sgn_sgn_rat,Xa) = $ite(
        Xa = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_if
tff(fact_5881_sgn__if,axiom,
    ! [Xa: code_integer] :
      sgn_sgn_Code_integer(Xa) = $ite(
        Xa = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Xa),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_5882_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int(A) = uminus_uminus_int(one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_5883_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_1_neg
tff(fact_5884_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_5885_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_5886_powr__non__neg,axiom,
    ! [A: real,Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa)),zero_zero_real) ).

% powr_non_neg
tff(fact_5887_powr__less__mono2__neg,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_5888_powr__ge__pzero,axiom,
    ! [Xa: 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,Xa),Ya)) ).

% powr_ge_pzero
tff(fact_5889_powr__mono2,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)) ) ) ) ).

% powr_mono2
tff(fact_5890_powr__less__cancel,axiom,
    ! [Xa: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% powr_less_cancel
tff(fact_5891_powr__less__mono,axiom,
    ! [A: real,B2: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),B2)) ) ) ).

% powr_less_mono
tff(fact_5892_powr__mono,axiom,
    ! [A: real,B2: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),B2)) ) ) ).

% powr_mono
tff(fact_5893_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,uminus_uminus_int(Z))),aa_int_nat(nat2,uminus_uminus_int(Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_5894_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_5895_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_5896_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_5897_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_5898_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_5899_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_5900_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_5901_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),uminus_uminus_int(one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_5902_zero__neq__neg__one,axiom,
    zero_zero_int != uminus_uminus_int(one_one_int) ).

% zero_neq_neg_one
tff(fact_5903_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5904_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5905_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5906_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5907_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int(A) = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5908_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5909_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B2 )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5910_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B2 )
    <=> ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5911_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 )
    <=> ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5912_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A = uminus_uminus_int(B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5913_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5914_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5915_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) )
    <=> ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5916_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5917_add_Oinverse__unique,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int )
     => ( uminus_uminus_int(A) = B2 ) ) ).

% add.inverse_unique
tff(fact_5918_add_Oinverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_5919_add_Oinverse__unique,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_5920_add_Oinverse__unique,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_5921_add_Oinverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_5922_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_5923_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5924_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5925_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_5926_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_5927_add__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int )
    <=> ( B2 = uminus_uminus_int(A) ) ) ).

% add_eq_0_iff
tff(fact_5928_add__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5929_add__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat )
    <=> ( B2 = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5930_add__eq__0__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger )
    <=> ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5931_add__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5932_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_5933_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_5934_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_5935_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_5936_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),uminus_uminus_int(one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_5937_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_5938_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_5939_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_5940_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5941_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5942_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5943_nonzero__minus__divide__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5944_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5945_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5946_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Na)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_5947_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),uminus_uminus_int(B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5948_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5949_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5950_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5951_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5952_diff__conv__add__uminus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),uminus_uminus_int(B2)) ).

% diff_conv_add_uminus
tff(fact_5953_diff__conv__add__uminus,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B2)) ).

% diff_conv_add_uminus
tff(fact_5954_diff__conv__add__uminus,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) ).

% diff_conv_add_uminus
tff(fact_5955_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% diff_conv_add_uminus
tff(fact_5956_diff__conv__add__uminus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% diff_conv_add_uminus
tff(fact_5957_group__cancel_Osub2,axiom,
    ! [B: int,K: int,B2: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_5958_group__cancel_Osub2,axiom,
    ! [B: real,K: real,B2: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),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,A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_5959_group__cancel_Osub2,axiom,
    ! [B: rat,K: rat,B2: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_5960_group__cancel_Osub2,axiom,
    ! [B: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
      ( ( B = plus_p5714425477246183910nteger(K,B2) )
     => ( minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B2)) ) ) ).

% group_cancel.sub2
tff(fact_5961_group__cancel_Osub2,axiom,
    ! [B: complex,K: complex,B2: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(K),B2) )
     => ( aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_5962_abs__leI,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2) ) ) ).

% abs_leI
tff(fact_5963_abs__leI,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2) ) ) ).

% abs_leI
tff(fact_5964_abs__leI,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2) ) ) ).

% abs_leI
tff(fact_5965_abs__leI,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2) ) ) ).

% abs_leI
tff(fact_5966_abs__le__D2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2) ) ).

% abs_le_D2
tff(fact_5967_abs__le__D2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) ) ).

% abs_le_D2
tff(fact_5968_abs__le__D2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ) ).

% abs_le_D2
tff(fact_5969_abs__le__D2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),B2) ) ).

% abs_le_D2
tff(fact_5970_abs__le__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5971_abs__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5972_abs__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5973_abs__le__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),B2) ) ) ).

% abs_le_iff
tff(fact_5974_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_5975_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_5976_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_5977_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_5978_abs__less__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(A)),B2) ) ) ).

% abs_less_iff
tff(fact_5979_abs__less__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5980_abs__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5981_abs__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5982_diff__eq,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,Xa),Ya) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),uminus6524753893492686040at_nat(Ya)) ).

% diff_eq
tff(fact_5983_diff__eq,axiom,
    ! [Xa: set_nat,Ya: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Xa),Ya) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),uminus5710092332889474511et_nat(Ya)) ).

% diff_eq
tff(fact_5984_inf__cancel__left1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(Xa)),B2)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left1
tff(fact_5985_inf__cancel__left1,axiom,
    ! [Xa: set_nat,A: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),B2)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_5986_inf__cancel__left1,axiom,
    ! [Xa: set_int,A: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(Xa)),B2)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_5987_inf__cancel__left1,axiom,
    ! [Xa: set_o,A: set_o,B2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(Xa)),B2)) = bot_bot_set_o ).

% inf_cancel_left1
tff(fact_5988_inf__cancel__left1,axiom,
    ! [Xa: set_set_nat,A: set_set_nat,B2: set_set_nat] : inf_inf_set_set_nat(inf_inf_set_set_nat(Xa,A),inf_inf_set_set_nat(uminus613421341184616069et_nat(Xa),B2)) = bot_bot_set_set_nat ).

% inf_cancel_left1
tff(fact_5989_inf__cancel__left1,axiom,
    ! [Xa: set_real,A: set_real,B2: set_real] : inf_inf_set_real(inf_inf_set_real(Xa,A),inf_inf_set_real(uminus612125837232591019t_real(Xa),B2)) = bot_bot_set_real ).

% inf_cancel_left1
tff(fact_5990_inf__cancel__left1,axiom,
    ! [Xa: set_Extended_enat,A: set_Extended_enat,B2: set_Extended_enat] : inf_in8357106775501769908d_enat(inf_in8357106775501769908d_enat(Xa,A),inf_in8357106775501769908d_enat(uminus417252749190364093d_enat(Xa),B2)) = bot_bo7653980558646680370d_enat ).

% inf_cancel_left1
tff(fact_5991_inf__cancel__left2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(Xa)),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),B2)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left2
tff(fact_5992_inf__cancel__left2,axiom,
    ! [Xa: set_nat,A: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(Xa)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),B2)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_5993_inf__cancel__left2,axiom,
    ! [Xa: set_int,A: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(Xa)),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),B2)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_5994_inf__cancel__left2,axiom,
    ! [Xa: set_o,A: set_o,B2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(Xa)),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),B2)) = bot_bot_set_o ).

% inf_cancel_left2
tff(fact_5995_inf__cancel__left2,axiom,
    ! [Xa: set_set_nat,A: set_set_nat,B2: set_set_nat] : inf_inf_set_set_nat(inf_inf_set_set_nat(uminus613421341184616069et_nat(Xa),A),inf_inf_set_set_nat(Xa,B2)) = bot_bot_set_set_nat ).

% inf_cancel_left2
tff(fact_5996_inf__cancel__left2,axiom,
    ! [Xa: set_real,A: set_real,B2: set_real] : inf_inf_set_real(inf_inf_set_real(uminus612125837232591019t_real(Xa),A),inf_inf_set_real(Xa,B2)) = bot_bot_set_real ).

% inf_cancel_left2
tff(fact_5997_inf__cancel__left2,axiom,
    ! [Xa: set_Extended_enat,A: set_Extended_enat,B2: set_Extended_enat] : inf_in8357106775501769908d_enat(inf_in8357106775501769908d_enat(uminus417252749190364093d_enat(Xa),A),inf_in8357106775501769908d_enat(Xa,B2)) = bot_bo7653980558646680370d_enat ).

% inf_cancel_left2
tff(fact_5998_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_5999_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_6000_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_6001_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_6002_mult__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M2)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) ).

% mult_Suc
tff(fact_6003_mult__eq__self__implies__10,axiom,
    ! [M2: nat,Na: nat] :
      ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) )
     => ( ( Na = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_6004_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_6005_not__int__zless__negative,axiom,
    ! [Na: nat,M2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,M2))) ).

% not_int_zless_negative
tff(fact_6006_div__times__less__eq__dividend,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),Na)),M2) ).

% div_times_less_eq_dividend
tff(fact_6007_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na))),M2) ).

% times_div_less_eq_dividend
tff(fact_6008_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)
     => 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(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_6009_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_6010_powr__less__mono2,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)) ) ) ) ).

% powr_less_mono2
tff(fact_6011_powr__mono2_H,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)) ) ) ) ).

% powr_mono2'
tff(fact_6012_gr__one__powr,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Ya)) ) ) ).

% gr_one_powr
tff(fact_6013_powr__inj,axiom,
    ! [A: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Ya) )
        <=> ( Xa = Ya ) ) ) ) ).

% powr_inj
tff(fact_6014_powr__le1,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_6015_powr__mono__both,axiom,
    ! [A: real,B2: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_6016_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_6017_powr__divide,axiom,
    ! [Xa: real,Ya: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Ya)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)) ) ) ) ).

% powr_divide
tff(fact_6018_powr__mult,axiom,
    ! [Xa: real,Ya: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Ya)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)) ) ) ) ).

% powr_mult
tff(fact_6019_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = one_one_Code_integer )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_6020_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_1_pos
tff(fact_6021_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_6022_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int(A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_6023_sgn__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) = aa_real_real(sgn_sgn_real,Xa) ) ) ).

% sgn_root
tff(fact_6024_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_6025_abs__sgn__eq,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% abs_sgn_eq
tff(fact_6026_abs__sgn__eq,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_6027_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,sgn_sgn_int(A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_6028_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_6029_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_6030_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_6031_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),uminus_uminus_int(one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_6032_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_6033_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_6034_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_6035_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_6036_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_6037_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_6038_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_6039_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_6040_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),uminus_uminus_int(one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_6041_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_6042_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_6043_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_6044_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6045_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6046_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) )
      <=> ( aa_complex_complex(times_times_complex(C2),B2) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6047_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A: real,C2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = C2 )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6048_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2)) = C2 )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6049_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A: complex,C2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = C2 )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C2),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6050_minus__divide__eq__eq,axiom,
    ! [B2: real,C2: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) = A )
    <=> $ite(C2 != zero_zero_real,aa_real_real(uminus_uminus_real,B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_6051_minus__divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)) = A )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_6052_minus__divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C2)) = A )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2) = aa_complex_complex(times_times_complex(A),C2),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_6053_eq__minus__divide__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( A = 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,A),C2) = aa_real_real(uminus_uminus_real,B2),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_6054_eq__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(uminus_uminus_rat,B2),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_6055_eq__minus__divide__eq,axiom,
    ! [A: complex,B2: complex,C2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C2)) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(uminus1482373934393186551omplex,B2),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_6056_divide__eq__minus__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B2 != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6057_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6058_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6059_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_6060_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6061_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6062_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6063_abs__eq__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(abs_abs_real,A) = B2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
        & ( ( A = B2 )
          | ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6064_abs__eq__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer(A) = B2 )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
        & ( ( A = B2 )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6065_abs__eq__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
        & ( ( A = B2 )
          | ( A = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6066_abs__eq__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(abs_abs_int,A) = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
        & ( ( A = B2 )
          | ( A = uminus_uminus_int(B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6067_eq__abs__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(abs_abs_real,B2) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B2 = A )
          | ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6068_eq__abs__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = abs_abs_Code_integer(B2) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B2 = A )
          | ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6069_eq__abs__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B2) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B2 = A )
          | ( B2 = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6070_eq__abs__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(abs_abs_int,B2) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B2 = A )
          | ( B2 = uminus_uminus_int(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6071_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = uminus_uminus_int(A) ) ) ).

% abs_of_neg
tff(fact_6072_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_6073_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_6074_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_6075_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),uminus_uminus_int(X4),X4) ).

% abs_if_raw
tff(fact_6076_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_6077_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_6078_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),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4),X4) ).

% abs_if_raw
tff(fact_6079_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),uminus_uminus_int(A),A) ).

% abs_if
tff(fact_6080_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_6081_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_6082_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_6083_inf__shunt,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya) = bot_bo2099793752762293965at_nat )
    <=> ord_le3146513528884898305at_nat(Xa,uminus6524753893492686040at_nat(Ya)) ) ).

% inf_shunt
tff(fact_6084_inf__shunt,axiom,
    ! [Xa: set_nat,Ya: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),uminus5710092332889474511et_nat(Ya)) ) ).

% inf_shunt
tff(fact_6085_inf__shunt,axiom,
    ! [Xa: set_o,Ya: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),uminus_uminus_set_o(Ya)) ) ).

% inf_shunt
tff(fact_6086_inf__shunt,axiom,
    ! [Xa: set_set_nat,Ya: set_set_nat] :
      ( ( inf_inf_set_set_nat(Xa,Ya) = bot_bot_set_set_nat )
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(Xa),uminus613421341184616069et_nat(Ya)) ) ).

% inf_shunt
tff(fact_6087_inf__shunt,axiom,
    ! [Xa: set_real,Ya: set_real] :
      ( ( inf_inf_set_real(Xa,Ya) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xa),uminus612125837232591019t_real(Ya)) ) ).

% inf_shunt
tff(fact_6088_inf__shunt,axiom,
    ! [Xa: set_Extended_enat,Ya: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(Xa,Ya) = bot_bo7653980558646680370d_enat )
    <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(Xa),uminus417252749190364093d_enat(Ya)) ) ).

% inf_shunt
tff(fact_6089_inf__shunt,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),uminus1532241313380277803et_int(Ya)) ) ).

% inf_shunt
tff(fact_6090_shunt1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),Ya),Z)
    <=> ord_le3146513528884898305at_nat(Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(Ya)),Z)) ) ).

% shunt1
tff(fact_6091_shunt1,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),Ya)),Z)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Ya)),Z)) ) ).

% shunt1
tff(fact_6092_shunt1,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),Ya)),Z)
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,uminus_uminus_set_o(Ya)),Z)) ) ).

% shunt1
tff(fact_6093_shunt1,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),Ya)),Z)
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(Ya)),Z)) ) ).

% shunt1
tff(fact_6094_shunt2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Ya: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Xa),uminus6524753893492686040at_nat(Ya)),Z)
    <=> ord_le3146513528884898305at_nat(Xa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Ya),Z)) ) ).

% shunt2
tff(fact_6095_shunt2,axiom,
    ! [Xa: set_nat,Ya: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Xa),uminus5710092332889474511et_nat(Ya))),Z)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ya),Z)) ) ).

% shunt2
tff(fact_6096_shunt2,axiom,
    ! [Xa: set_o,Ya: set_o,Z: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Xa),uminus_uminus_set_o(Ya))),Z)
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ya),Z)) ) ).

% shunt2
tff(fact_6097_shunt2,axiom,
    ! [Xa: set_int,Ya: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Xa),uminus1532241313380277803et_int(Ya))),Z)
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ya),Z)) ) ).

% shunt2
tff(fact_6098_sup__neg__inf,axiom,
    ! [P3: set_Pr1261947904930325089at_nat,Q5: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(P3,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q5),R2))
    <=> ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P3),uminus6524753893492686040at_nat(Q5)),R2) ) ).

% sup_neg_inf
tff(fact_6099_sup__neg__inf,axiom,
    ! [P3: set_nat,Q5: set_nat,R2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,P3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q5),R2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P3),uminus5710092332889474511et_nat(Q5))),R2) ) ).

% sup_neg_inf
tff(fact_6100_sup__neg__inf,axiom,
    ! [P3: set_o,Q5: set_o,R2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(P3),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Q5),R2))
    <=> aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,P3),uminus_uminus_set_o(Q5))),R2) ) ).

% sup_neg_inf
tff(fact_6101_sup__neg__inf,axiom,
    ! [P3: set_int,Q5: set_int,R2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(P3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Q5),R2))
    <=> aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,P3),uminus1532241313380277803et_int(Q5))),R2) ) ).

% sup_neg_inf
tff(fact_6102_n__less__n__mult__m,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M2)) ) ) ).

% n_less_n_mult_m
tff(fact_6103_n__less__m__mult__n,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_6104_one__less__mult,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2)
       => 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,M2),Na)) ) ) ).

% one_less_mult
tff(fact_6105_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N2: nat] : M2 != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( M2 != uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_6106_int__zle__neg,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,M2)))
    <=> ( ( Na = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_6107_div__less__iff__less__mult,axiom,
    ! [Q5: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Q5)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)) ) ) ).

% div_less_iff_less_mult
tff(fact_6108_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_6109_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N2: nat] : K != uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_6110_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),uminus_uminus_int(I),I) ).

% zabs_def
tff(fact_6111_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M2)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_6112_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_6113_powr__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na) ) ) ).

% powr_realpow
tff(fact_6114_powr__less__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_6115_less__powr__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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),Xa)),Ya) ) ) ) ).

% less_powr_iff
tff(fact_6116_log__less__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)),Ya)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)) ) ) ) ).

% log_less_iff
tff(fact_6117_less__log__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xa) ) ) ) ).

% less_log_iff
tff(fact_6118_less__minus__divide__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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,A),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,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_6119_less__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_6120_minus__divide__less__eq,axiom,
    ! [B2: real,C2: real,A: 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))),A)
    <=> $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,A),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,A),C2)),aa_real_real(uminus_uminus_real,B2)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6121_minus__divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))),A)
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6122_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6123_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6124_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B2: real,A: 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))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6125_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6126_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6127_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6128_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B2: real,A: 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))),A)
      <=> 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,A),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6129_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6130_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xa: 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,Xa),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,Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6131_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z))),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6132_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xa),Z))),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xa)),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6133_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: 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,A),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,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6134_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B2) = $ite(Z = zero_zero_rat,B2,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6135_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B2: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B2) = $ite(Z = zero_zero_complex,B2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6136_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xa: 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,Xa),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,Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6137_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Xa),Z))),Ya) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Ya),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6138_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xa),Z))),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xa)),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6139_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),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,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6140_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B2) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6141_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B2: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B2) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6142_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: 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,A),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,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6143_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B2) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6144_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B2: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),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,A)),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6145_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) )
       => ~ ! [N2: nat] :
              ( ( K = uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% int_cases3
tff(fact_6146_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))) ).

% not_zle_0_negative
tff(fact_6147_negative__zless__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_6148_negD,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
     => ? [N2: nat] : Xa = uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6149_div__nat__eqI,axiom,
    ! [Na: nat,Q5: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q5)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = Q5 ) ) ) ).

% div_nat_eqI
tff(fact_6150_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q5: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Q5))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q5)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_6151_split__div,axiom,
    ! [P: fun_nat_o,M2: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I4)),J3) )
               => aa_nat_o(P,I4) ) ) ) ) ) ).

% split_div
tff(fact_6152_dividend__less__div__times,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_6153_dividend__less__times__div,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)))) ) ).

% dividend_less_times_div
tff(fact_6154_mult__eq__if,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na) = $ite(M2 = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),Na))) ).

% mult_eq_if
tff(fact_6155_verit__less__mono__div__int2,axiom,
    ! [A2: int,B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),uminus_uminus_int(Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_6156_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,uminus_uminus_int(one_one_int)),B2) = uminus_uminus_int(one_one_int) ) ) ).

% div_eq_minus1
tff(fact_6157_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_6158_powr__mult__base,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Ya)) ) ) ).

% powr_mult_base
tff(fact_6159_sgn__power__injE,axiom,
    ! [A: real,Na: nat,Xa: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) = Xa )
     => ( ( Xa = aa_real_real(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)),Na)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( A = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_6160_le__log__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xa) ) ) ) ).

% le_log_iff
tff(fact_6161_log__le__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)),Ya)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)) ) ) ) ).

% log_le_iff
tff(fact_6162_le__powr__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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),Xa)),Ya) ) ) ) ).

% le_powr_iff
tff(fact_6163_powr__le__iff,axiom,
    ! [B2: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_6164_le__minus__divide__eq,axiom,
    ! [A: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),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,A),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_6165_le__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_6166_minus__divide__le__eq,axiom,
    ! [B2: real,C2: real,A: 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))),A)
    <=> $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,A),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,A),C2)),aa_real_real(uminus_uminus_real,B2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6167_minus__divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))),A)
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,A),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6168_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6169_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6170_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B2: real,A: 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))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6171_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6172_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A: 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,A),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,A),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6173_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: 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,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6174_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B2: real,A: 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))),A)
      <=> 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,A),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6175_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6176_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6177_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6178_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6179_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N2: nat] :
            ( ( K = uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% neg_int_cases
tff(fact_6180_split__div_H,axiom,
    ! [P: fun_nat_o,M2: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q6)),M2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q6)))
            & aa_nat_o(P,Q6) ) ) ) ).

% split_div'
tff(fact_6181_ln__powr__bound,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_6182_ln__powr__bound2,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xa)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_6183_add__log__eq__powr,axiom,
    ! [B2: real,Xa: 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),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) = 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)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_6184_log__add__eq__powr,axiom,
    ! [B2: real,Xa: 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),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_6185_minus__log__eq__powr,axiom,
    ! [B2: real,Xa: 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),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) = 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)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_6186_root__sgn__power,axiom,
    ! [Na: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(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)),Na))) = Ya ) ) ).

% root_sgn_power
tff(fact_6187_sgn__power__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))),Na)) = Xa ) ) ).

% sgn_power_root
tff(fact_6188_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_6189_sgn__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_6190_zero__le__sgn__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_sgn_iff
tff(fact_6191_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_6192_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_6193_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(power_power_int(uminus_uminus_int(one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_int(power_power_int(uminus_uminus_int(one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6194_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6195_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6196_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6197_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6198_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_6199_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M2)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_6200_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B))
    <=> aa_set_int_o(ord_less_eq_set_int(B),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_6201_Compl__anti__mono,axiom,
    ! [A2: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(B)),uminus1532241313380277803et_int(A2)) ) ).

% Compl_anti_mono
tff(fact_6202_Compl__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),uminus6524753893492686040at_nat(A2)) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint
tff(fact_6203_Compl__disjoint,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),uminus5710092332889474511et_nat(A2)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_6204_Compl__disjoint,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),uminus1532241313380277803et_int(A2)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_6205_Compl__disjoint,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),uminus_uminus_set_o(A2)) = bot_bot_set_o ).

% Compl_disjoint
tff(fact_6206_Compl__disjoint,axiom,
    ! [A2: set_set_nat] : inf_inf_set_set_nat(A2,uminus613421341184616069et_nat(A2)) = bot_bot_set_set_nat ).

% Compl_disjoint
tff(fact_6207_Compl__disjoint,axiom,
    ! [A2: set_real] : inf_inf_set_real(A2,uminus612125837232591019t_real(A2)) = bot_bot_set_real ).

% Compl_disjoint
tff(fact_6208_Compl__disjoint,axiom,
    ! [A2: set_Extended_enat] : inf_in8357106775501769908d_enat(A2,uminus417252749190364093d_enat(A2)) = bot_bo7653980558646680370d_enat ).

% Compl_disjoint
tff(fact_6209_Compl__disjoint2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A2)),A2) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint2
tff(fact_6210_Compl__disjoint2,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A2)),A2) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_6211_Compl__disjoint2,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(A2)),A2) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_6212_Compl__disjoint2,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(A2)),A2) = bot_bot_set_o ).

% Compl_disjoint2
tff(fact_6213_Compl__disjoint2,axiom,
    ! [A2: set_set_nat] : inf_inf_set_set_nat(uminus613421341184616069et_nat(A2),A2) = bot_bot_set_set_nat ).

% Compl_disjoint2
tff(fact_6214_Compl__disjoint2,axiom,
    ! [A2: set_real] : inf_inf_set_real(uminus612125837232591019t_real(A2),A2) = bot_bot_set_real ).

% Compl_disjoint2
tff(fact_6215_Compl__disjoint2,axiom,
    ! [A2: set_Extended_enat] : inf_in8357106775501769908d_enat(uminus417252749190364093d_enat(A2),A2) = bot_bo7653980558646680370d_enat ).

% Compl_disjoint2
tff(fact_6216_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_6217_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_6218_Diff__Compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),uminus6524753893492686040at_nat(B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) ).

% Diff_Compl
tff(fact_6219_Diff__Compl,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),uminus5710092332889474511et_nat(B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) ).

% Diff_Compl
tff(fact_6220_Compl__Diff__eq,axiom,
    ! [A2: set_o,B: set_o] : uminus_uminus_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,A2),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,uminus_uminus_set_o(A2)),B) ).

% Compl_Diff_eq
tff(fact_6221_Compl__Diff__eq,axiom,
    ! [A2: set_int,B: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(A2)),B) ).

% Compl_Diff_eq
tff(fact_6222_Compl__Diff__eq,axiom,
    ! [A2: set_nat,B: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A2)),B) ).

% Compl_Diff_eq
tff(fact_6223_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(insert_nat(B2,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B2),A2) ) ).

% subset_Compl_singleton
tff(fact_6224_subset__Compl__singleton,axiom,
    ! [A2: set_o,B2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(insert_o((B2),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B2)),A2) ) ).

% subset_Compl_singleton
tff(fact_6225_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B2: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B2,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B2),A2) ) ).

% subset_Compl_singleton
tff(fact_6226_subset__Compl__singleton,axiom,
    ! [A2: set_real,B2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B2,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B2),A2) ) ).

% subset_Compl_singleton
tff(fact_6227_subset__Compl__singleton,axiom,
    ! [A2: set_Extended_enat,B2: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),uminus417252749190364093d_enat(insert_Extended_enat(B2,bot_bo7653980558646680370d_enat)))
    <=> ~ aa_set3337841544853333262enat_o(member_Extended_enat(B2),A2) ) ).

% subset_Compl_singleton
tff(fact_6228_subset__Compl__singleton,axiom,
    ! [A2: set_int,B2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B2,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B2),A2) ) ).

% subset_Compl_singleton
tff(fact_6229_artanh__minus__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xa)) ) ) ).

% artanh_minus_real
tff(fact_6230_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6231_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(A2))
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_6232_subset__Compl__self__eq,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(A2))
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6233_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_6234_subset__Compl__self__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),uminus417252749190364093d_enat(A2))
    <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% subset_Compl_self_eq
tff(fact_6235_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_6236_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_6237_Compl__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A2)),uminus6524753893492686040at_nat(B)) ).

% Compl_Un
tff(fact_6238_Compl__Un,axiom,
    ! [A2: set_nat,B: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B)) ).

% Compl_Un
tff(fact_6239_Compl__Un,axiom,
    ! [A2: set_o,B: set_o] : uminus_uminus_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,uminus_uminus_set_o(A2)),uminus_uminus_set_o(B)) ).

% Compl_Un
tff(fact_6240_Compl__Un,axiom,
    ! [A2: set_int,B: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B)) ).

% Compl_Un
tff(fact_6241_Compl__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(A2)),uminus6524753893492686040at_nat(B)) ).

% Compl_Int
tff(fact_6242_Compl__Int,axiom,
    ! [A2: set_nat,B: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B)) ).

% Compl_Int
tff(fact_6243_Compl__Int,axiom,
    ! [A2: set_o,B: set_o] : uminus_uminus_set_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,uminus_uminus_set_o(A2)),uminus_uminus_set_o(B)) ).

% Compl_Int
tff(fact_6244_Compl__Int,axiom,
    ! [A2: set_int,B: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B)) ).

% Compl_Int
tff(fact_6245_Diff__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),uminus6524753893492686040at_nat(B)) ).

% Diff_eq
tff(fact_6246_Diff__eq,axiom,
    ! [A2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),uminus5710092332889474511et_nat(B)) ).

% Diff_eq
tff(fact_6247_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = bot_bo2099793752762293965at_nat )
    <=> ord_le3146513528884898305at_nat(A2,uminus6524753893492686040at_nat(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6248_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6249_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6250_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( inf_inf_set_set_nat(A2,B) = bot_bot_set_set_nat )
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6251_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( inf_inf_set_real(A2,B) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6252_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( inf_in8357106775501769908d_enat(A2,B) = bot_bo7653980558646680370d_enat )
    <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),uminus417252749190364093d_enat(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6253_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(B)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6254_Compl__insert,axiom,
    ! [Xa: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(Xa,A2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,uminus1532241313380277803et_int(A2)),insert_int(Xa,bot_bot_set_int)) ).

% Compl_insert
tff(fact_6255_Compl__insert,axiom,
    ! [Xa: $o,A2: set_o] : uminus_uminus_set_o(insert_o((Xa),A2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,uminus_uminus_set_o(A2)),insert_o((Xa),bot_bot_set_o)) ).

% Compl_insert
tff(fact_6256_Compl__insert,axiom,
    ! [Xa: set_nat,A2: set_set_nat] : uminus613421341184616069et_nat(insert_set_nat(Xa,A2)) = minus_2163939370556025621et_nat(uminus613421341184616069et_nat(A2),insert_set_nat(Xa,bot_bot_set_set_nat)) ).

% Compl_insert
tff(fact_6257_Compl__insert,axiom,
    ! [Xa: real,A2: set_real] : uminus612125837232591019t_real(insert_real(Xa,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(Xa,bot_bot_set_real)) ).

% Compl_insert
tff(fact_6258_Compl__insert,axiom,
    ! [Xa: extended_enat,A2: set_Extended_enat] : uminus417252749190364093d_enat(insert_Extended_enat(Xa,A2)) = minus_925952699566721837d_enat(uminus417252749190364093d_enat(A2),insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) ).

% Compl_insert
tff(fact_6259_Compl__insert,axiom,
    ! [Xa: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(Xa,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,uminus5710092332889474511et_nat(A2)),insert_nat(Xa,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_6260_real__add__less__0__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_less_0_iff
tff(fact_6261_real__0__less__add__iff,axiom,
    ! [Xa: 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,Xa),Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xa)),Ya) ) ).

% real_0_less_add_iff
tff(fact_6262_real__0__le__add__iff,axiom,
    ! [Xa: 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,Xa),Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),Ya) ) ).

% real_0_le_add_iff
tff(fact_6263_real__add__le__0__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_le_0_iff
tff(fact_6264_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_6265_zsgn__def,axiom,
    ! [I: int] :
      sgn_sgn_int(I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,uminus_uminus_int(one_one_int)) ) ).

% zsgn_def
tff(fact_6266_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_6267_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xa) ) ) ).

% powr_neg_one
tff(fact_6268_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_6269_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xa))),aa_real_real(uminus_uminus_real,Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6270_Bernoulli__inequality,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Na)) ) ).

% Bernoulli_inequality
tff(fact_6271_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M2 = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_6272_sgn__zero__iff,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,Xa) = zero_zero_complex )
    <=> ( Xa = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_6273_sgn__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sgn_sgn_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_6274_log__minus__eq__powr,axiom,
    ! [B2: real,Xa: 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),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_real_real(uminus_uminus_real,Ya)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_6275_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( M2 = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_6276_nat__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_6277_nat__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_6278_nat__mult__div__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_6279_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_6280_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M2)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_6281_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_6282_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M2)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_6283_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_6284_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M2) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_6285_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_nat_real(semiri5074537144036343181t_real,K))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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_6286_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,uminus_uminus_int(I))))) ) ) ).

% powr_int
tff(fact_6287_dbl__dec__simps_I2_J,axiom,
    neg_nu3811975205180677377ec_int(zero_zero_int) = uminus_uminus_int(one_one_int) ).

% dbl_dec_simps(2)
tff(fact_6288_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_6289_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_6290_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6291_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_6292_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_6293_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_6294_Gcd__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( gcd_Gcd_nat(A2) = zero_zero_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(zero_zero_nat,bot_bot_set_nat)) ) ).

% Gcd_0_iff
tff(fact_6295_Gcd__0__iff,axiom,
    ! [A2: set_int] :
      ( ( gcd_Gcd_int(A2) = zero_zero_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),insert_int(zero_zero_int,bot_bot_set_int)) ) ).

% Gcd_0_iff
tff(fact_6296_ComplI,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2)
     => aa_set3337841544853333262enat_o(member_Extended_enat(C2),uminus417252749190364093d_enat(A2)) ) ).

% ComplI
tff(fact_6297_ComplI,axiom,
    ! [C2: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(C2),A2)
     => aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A2)) ) ).

% ComplI
tff(fact_6298_ComplI,axiom,
    ! [C2: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(C2),A2)
     => aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A2)) ) ).

% ComplI
tff(fact_6299_ComplI,axiom,
    ! [C2: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C2),A2)
     => aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A2)) ) ).

% ComplI
tff(fact_6300_ComplI,axiom,
    ! [C2: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(C2),A2)
     => aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A2)) ) ).

% ComplI
tff(fact_6301_ComplI,axiom,
    ! [C2: $o,A2: set_o] :
      ( ~ aa_set_o_o(member_o((C2)),A2)
     => aa_set_o_o(member_o((C2)),uminus_uminus_set_o(A2)) ) ).

% ComplI
tff(fact_6302_Compl__iff,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),uminus417252749190364093d_enat(A2))
    <=> ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2) ) ).

% Compl_iff
tff(fact_6303_Compl__iff,axiom,
    ! [C2: real,A2: set_real] :
      ( aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A2))
    <=> ~ aa_set_real_o(member_real(C2),A2) ) ).

% Compl_iff
tff(fact_6304_Compl__iff,axiom,
    ! [C2: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A2))
    <=> ~ aa_set_set_nat_o(member_set_nat(C2),A2) ) ).

% Compl_iff
tff(fact_6305_Compl__iff,axiom,
    ! [C2: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A2))
    <=> ~ aa_set_nat_o(member_nat(C2),A2) ) ).

% Compl_iff
tff(fact_6306_Compl__iff,axiom,
    ! [C2: int,A2: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A2))
    <=> ~ aa_set_int_o(member_int(C2),A2) ) ).

% Compl_iff
tff(fact_6307_Compl__iff,axiom,
    ! [C2: $o,A2: set_o] :
      ( aa_set_o_o(member_o((C2)),uminus_uminus_set_o(A2))
    <=> ~ aa_set_o_o(member_o((C2)),A2) ) ).

% Compl_iff
tff(fact_6308_exp__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Ya)) ) ).

% exp_less_mono
tff(fact_6309_exp__less__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% exp_less_cancel_iff
tff(fact_6310_exp__le__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% exp_le_cancel_iff
tff(fact_6311_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_6312_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_6313_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_6314_dbl__dec__simps_I3_J,axiom,
    neg_nu3811975205180677377ec_int(one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_6315_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_int(Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_6316_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_6317_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_6318_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = ring_1_of_int_int(Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_6319_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_6320_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_6321_of__int__0,axiom,
    ring_1_of_int_int(zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_6322_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_6323_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_6324_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_6325_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_6326_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6327_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6328_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(W2)),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6329_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6330_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6331_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(W2)),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6332_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_6333_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_6334_Gcd__empty,axiom,
    gcd_Gcd_nat(bot_bot_set_nat) = zero_zero_nat ).

% Gcd_empty
tff(fact_6335_Gcd__empty,axiom,
    gcd_Gcd_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_empty
tff(fact_6336_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% one_less_exp_iff
tff(fact_6337_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_6338_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_6339_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% one_le_exp_iff
tff(fact_6340_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% exp_ln_iff
tff(fact_6341_exp__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa ) ) ).

% exp_ln
tff(fact_6342_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_6343_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_6344_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6345_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_6346_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_6347_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6348_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_6349_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_6350_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6351_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_6352_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_6353_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6354_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_6355_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_6356_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6357_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_6358_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_6359_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6360_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_6361_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_6362_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6363_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_6364_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_6365_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6366_ceiling__le__zero,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_6367_ceiling__le__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_6368_zero__less__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa) ) ).

% zero_less_ceiling
tff(fact_6369_zero__less__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% zero_less_ceiling
tff(fact_6370_ceiling__less__one,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_6371_ceiling__less__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_6372_one__le__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa) ) ).

% one_le_ceiling
tff(fact_6373_one__le__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% one_le_ceiling
tff(fact_6374_ceiling__le__one,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ).

% ceiling_le_one
tff(fact_6375_ceiling__le__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat) ) ).

% ceiling_le_one
tff(fact_6376_one__less__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ).

% one_less_ceiling
tff(fact_6377_one__less__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ).

% one_less_ceiling
tff(fact_6378_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,Xa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W2)),aa_int_real(ring_1_of_int_real,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W2)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6379_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W2)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W2)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6380_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(ring_1_of_int_int(B2)),W2)),ring_1_of_int_int(Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W2)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6381_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B2: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B2),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6382_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B2: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xa)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B2),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6383_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B2: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(Xa)),aa_nat_int(power_power_int(ring_1_of_int_int(B2)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B2),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6384_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,Xa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W2)),aa_int_real(ring_1_of_int_real,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W2)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6385_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W2)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W2)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6386_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(ring_1_of_int_int(B2)),W2)),ring_1_of_int_int(Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W2)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6387_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B2: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B2),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6388_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B2: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xa)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B2),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6389_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B2: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(Xa)),aa_nat_int(power_power_int(ring_1_of_int_int(B2)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B2),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6390_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).

% of_nat_nat
tff(fact_6391_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = ring_1_of_int_int(Z) ) ) ).

% of_nat_nat
tff(fact_6392_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_6393_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_6394_ceiling__less__zero,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_6395_ceiling__less__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_6396_zero__le__ceiling,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa) ) ).

% zero_le_ceiling
tff(fact_6397_zero__le__ceiling,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa) ) ).

% zero_le_ceiling
tff(fact_6398_ComplD,axiom,
    ! [C2: extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(C2),uminus417252749190364093d_enat(A2))
     => ~ aa_set3337841544853333262enat_o(member_Extended_enat(C2),A2) ) ).

% ComplD
tff(fact_6399_ComplD,axiom,
    ! [C2: real,A2: set_real] :
      ( aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A2))
     => ~ aa_set_real_o(member_real(C2),A2) ) ).

% ComplD
tff(fact_6400_ComplD,axiom,
    ! [C2: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A2))
     => ~ aa_set_set_nat_o(member_set_nat(C2),A2) ) ).

% ComplD
tff(fact_6401_ComplD,axiom,
    ! [C2: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A2))
     => ~ aa_set_nat_o(member_nat(C2),A2) ) ).

% ComplD
tff(fact_6402_ComplD,axiom,
    ! [C2: int,A2: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A2))
     => ~ aa_set_int_o(member_int(C2),A2) ) ).

% ComplD
tff(fact_6403_ComplD,axiom,
    ! [C2: $o,A2: set_o] :
      ( aa_set_o_o(member_o((C2)),uminus_uminus_set_o(A2))
     => ~ aa_set_o_o(member_o((C2)),A2) ) ).

% ComplD
tff(fact_6404_le__of__int__ceiling,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xa))) ).

% le_of_int_ceiling
tff(fact_6405_le__of__int__ceiling,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa))) ).

% le_of_int_ceiling
tff(fact_6406_ceiling__le,axiom,
    ! [Xa: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),A) ) ).

% ceiling_le
tff(fact_6407_ceiling__le,axiom,
    ! [Xa: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),A) ) ).

% ceiling_le
tff(fact_6408_ceiling__le__iff,axiom,
    ! [Xa: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_6409_ceiling__le__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_6410_less__ceiling__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xa) ) ).

% less_ceiling_iff
tff(fact_6411_less__ceiling__iff,axiom,
    ! [Z: int,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xa) ) ).

% less_ceiling_iff
tff(fact_6412_ex__le__of__int,axiom,
    ! [Xa: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_6413_ex__le__of__int,axiom,
    ! [Xa: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_6414_exp__less__cancel,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Ya))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% exp_less_cancel
tff(fact_6415_ex__of__int__less,axiom,
    ! [Xa: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xa) ).

% ex_of_int_less
tff(fact_6416_ex__of__int__less,axiom,
    ! [Xa: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xa) ).

% ex_of_int_less
tff(fact_6417_ex__less__of__int,axiom,
    ! [Xa: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_6418_ex__less__of__int,axiom,
    ! [Xa: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_6419_exp__not__eq__zero,axiom,
    ! [Xa: real] : aa_real_real(exp_real,Xa) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_6420_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_6421_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_6422_of__int__ceiling__diff__one__le,axiom,
    ! [R2: 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,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_6423_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_6424_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I4: 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,I4)),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,I4)) )
         => aa_int_o(P,I4) ) ) ).

% ceiling_split
tff(fact_6425_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,Ta))
    <=> ! [I4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,I4)),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,I4)) )
         => aa_int_o(P,I4) ) ) ).

% ceiling_split
tff(fact_6426_ceiling__eq__iff,axiom,
    ! [Xa: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xa) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6427_ceiling__eq__iff,axiom,
    ! [Xa: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,Xa) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6428_ceiling__unique,axiom,
    ! [Z: int,Xa: 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)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xa) = Z ) ) ) ).

% ceiling_unique
tff(fact_6429_ceiling__unique,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xa) = Z ) ) ) ).

% ceiling_unique
tff(fact_6430_ceiling__correct,axiom,
    ! [Xa: 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,Xa))),one_one_real)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xa))) ) ).

% ceiling_correct
tff(fact_6431_ceiling__correct,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa))),one_one_rat)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xa))) ) ).

% ceiling_correct
tff(fact_6432_ceiling__less__iff,axiom,
    ! [Xa: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_6433_ceiling__less__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_6434_le__ceiling__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xa) ) ).

% le_ceiling_iff
tff(fact_6435_le__ceiling__iff,axiom,
    ! [Z: int,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xa) ) ).

% le_ceiling_iff
tff(fact_6436_exp__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ? [X2: real] : aa_real_real(exp_real,X2) = Ya ) ).

% exp_total
tff(fact_6437_exp__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_gt_zero
tff(fact_6438_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_6439_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_6440_exp__ge__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_ge_zero
tff(fact_6441_ceiling__mono,axiom,
    ! [Ya: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Ya)),aa_real_int(archim7802044766580827645g_real,Xa)) ) ).

% ceiling_mono
tff(fact_6442_ceiling__mono,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Ya)),aa_rat_int(archim2889992004027027881ng_rat,Xa)) ) ).

% ceiling_mono
tff(fact_6443_ceiling__less__cancel,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Ya))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya) ) ).

% ceiling_less_cancel
tff(fact_6444_ceiling__less__cancel,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_real_int(archim7802044766580827645g_real,Ya))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% ceiling_less_cancel
tff(fact_6445_ceiling__divide__upper,axiom,
    ! [Q5: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q5)
     => 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),Q5)))),Q5)) ) ).

% ceiling_divide_upper
tff(fact_6446_ceiling__divide__upper,axiom,
    ! [Q5: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q5)))),Q5)) ) ).

% ceiling_divide_upper
tff(fact_6447_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K4)) ).

% Gcd_int_greater_eq_0
tff(fact_6448_ceiling__divide__lower,axiom,
    ! [Q5: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q5)))),one_one_rat)),Q5)),P3) ) ).

% ceiling_divide_lower
tff(fact_6449_ceiling__divide__lower,axiom,
    ! [Q5: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q5)
     => 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),Q5)))),one_one_real)),Q5)),P3) ) ).

% ceiling_divide_lower
tff(fact_6450_ceiling__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6451_ceiling__eq,axiom,
    ! [Na: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Na)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Na)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6452_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa)) ) ).

% exp_gt_one
tff(fact_6453_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ).

% exp_ge_add_one_self
tff(fact_6454_of__nat__ceiling,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2)))) ).

% of_nat_ceiling
tff(fact_6455_of__nat__ceiling,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R2)))) ).

% of_nat_ceiling
tff(fact_6456_ceiling__add__le,axiom,
    ! [Xa: rat,Ya: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Ya))) ).

% ceiling_add_le
tff(fact_6457_ceiling__add__le,axiom,
    ! [Xa: 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,Xa),Ya))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_real_int(archim7802044766580827645g_real,Ya))) ).

% ceiling_add_le
tff(fact_6458_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_6459_real__of__int__div4,axiom,
    ! [Na: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xa))) ).

% real_of_int_div4
tff(fact_6460_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_6461_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_6462_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_6463_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),ring_1_of_int_int(Z)) ) ).

% of_int_nonneg
tff(fact_6464_of__int__leD,axiom,
    ! [Na: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_leD
tff(fact_6465_of__int__leD,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% of_int_leD
tff(fact_6466_of__int__leD,axiom,
    ! [Na: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ) ).

% of_int_leD
tff(fact_6467_of__int__leD,axiom,
    ! [Na: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,ring_1_of_int_int(Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xa) ) ) ).

% of_int_leD
tff(fact_6468_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_6469_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_6470_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),ring_1_of_int_int(Z)) ) ).

% of_int_pos
tff(fact_6471_of__int__lessD,axiom,
    ! [Na: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_lessD
tff(fact_6472_of__int__lessD,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% of_int_lessD
tff(fact_6473_of__int__lessD,axiom,
    ! [Na: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ) ).

% of_int_lessD
tff(fact_6474_of__int__lessD,axiom,
    ! [Na: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,ring_1_of_int_int(Na))),Xa)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xa) ) ) ).

% of_int_lessD
tff(fact_6475_lemma__exp__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Ya)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),one_one_real))
          & ( aa_real_real(exp_real,X2) = Ya ) ) ) ).

% lemma_exp_total
tff(fact_6476_ln__ge__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Ya)),Xa) ) ) ).

% ln_ge_iff
tff(fact_6477_floor__exists1,axiom,
    ! [Xa: real] :
    ? [X2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y2)),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y2),one_one_int))) )
         => ( Y2 = X2 ) ) ) ).

% floor_exists1
tff(fact_6478_floor__exists1,axiom,
    ! [Xa: rat] :
    ? [X2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y2)),Xa)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y2),one_one_int))) )
         => ( Y2 = X2 ) ) ) ).

% floor_exists1
tff(fact_6479_floor__exists,axiom,
    ! [Xa: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_6480_floor__exists,axiom,
    ! [Xa: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_6481_ln__x__over__x__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_6482_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_6483_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),ring_1_of_int_int(Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_6484_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_int_real(ring_1_of_int_real,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xa) ) ).

% of_nat_less_of_int_iff
tff(fact_6485_int__le__real__less,axiom,
    ! [Na: int,M2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M2)),one_one_real)) ) ).

% int_le_real_less
tff(fact_6486_int__less__real__le,axiom,
    ! [Na: int,M2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),M2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real)),aa_int_real(ring_1_of_int_real,M2)) ) ).

% int_less_real_le
tff(fact_6487_powr__def,axiom,
    ! [Xa: real,A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A) = $ite(Xa = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,Xa)))) ).

% powr_def
tff(fact_6488_mult__ceiling__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,A),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2))) ) ) ).

% mult_ceiling_le
tff(fact_6489_mult__ceiling__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2))) ) ) ).

% mult_ceiling_le
tff(fact_6490_exp__divide__power__eq,axiom,
    ! [Na: nat,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xa),aa_nat_complex(semiri8010041392384452111omplex,Na)))),Na) = aa_complex_complex(exp_complex,Xa) ) ) ).

% exp_divide_power_eq
tff(fact_6491_exp__divide__power__eq,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na) = aa_real_real(exp_real,Xa) ) ) ).

% exp_divide_power_eq
tff(fact_6492_real__of__int__div2,axiom,
    ! [Na: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa)))) ).

% real_of_int_div2
tff(fact_6493_real__of__int__div3,axiom,
    ! [Na: int,Xa: 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,Na)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa)))),one_one_real) ).

% real_of_int_div3
tff(fact_6494_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M5)
     => ( gcd_Gcd_nat(M5) = gcd_Gcd_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M5),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_6495_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,uminus_uminus_int(K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6496_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),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,semiri4939895301339042750nteger(aa_int_nat(nat2,uminus_uminus_int(K)))),semiri4939895301339042750nteger(aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6497_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,uminus_uminus_int(K)))),aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6498_of__int__of__nat,axiom,
    ! [K: int] :
      ring_1_of_int_int(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),uminus_uminus_int(aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,uminus_uminus_int(K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6499_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,uminus_uminus_int(K)))),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_6500_dbl__dec__def,axiom,
    ! [Xa: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xa) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xa),Xa)),one_one_complex) ).

% dbl_dec_def
tff(fact_6501_dbl__dec__def,axiom,
    ! [Xa: real] : aa_real_real(neg_nu6075765906172075777c_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa)),one_one_real) ).

% dbl_dec_def
tff(fact_6502_dbl__dec__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xa) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa)),one_one_rat) ).

% dbl_dec_def
tff(fact_6503_dbl__dec__def,axiom,
    ! [Xa: int] : neg_nu3811975205180677377ec_int(Xa) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa)),one_one_int) ).

% dbl_dec_def
tff(fact_6504_powr__real__of__int,axiom,
    ! [Xa: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_int_real(ring_1_of_int_real,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,Na)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,uminus_uminus_int(Na))))) ) ) ).

% powr_real_of_int
tff(fact_6505_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xa)) = 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))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_6506_sinh__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sinh_real,Xa) = zero_zero_real )
    <=> aa_set_real_o(member_real(aa_real_real(exp_real,Xa)),insert_real(one_one_real,insert_real(aa_real_real(uminus_uminus_real,one_one_real),bot_bot_set_real))) ) ).

% sinh_zero_iff
tff(fact_6507_sinh__zero__iff,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(sinh_complex,Xa) = zero_zero_complex )
    <=> aa_set_complex_o(member_complex(aa_complex_complex(exp_complex,Xa)),insert_complex(one_one_complex,insert_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex),bot_bot_set_complex))) ) ).

% sinh_zero_iff
tff(fact_6508_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_set_real_o(member_real(A),ring_1_Ints_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,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6509_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_set_real_o(member_real(A),ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6510_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_set_real_o(member_real(A),ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))),ring_1_of_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6511_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6512_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_set_rat_o(member_rat(A),ring_1_Ints_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,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6513_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))),ring_1_of_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6514_floor__divide__upper,axiom,
    ! [Q5: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q5)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q5)))),one_one_real)),Q5)) ) ).

% floor_divide_upper
tff(fact_6515_floor__divide__upper,axiom,
    ! [Q5: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q5)))),one_one_rat)),Q5)) ) ).

% floor_divide_upper
tff(fact_6516_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat)
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_6517_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => ( rotate1_o(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_6518_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_6519_sinh__real__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),aa_real_real(sinh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% sinh_real_less_iff
tff(fact_6520_sinh__real__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),aa_real_real(sinh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% sinh_real_le_iff
tff(fact_6521_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% sinh_real_pos_iff
tff(fact_6522_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_6523_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_6524_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_6525_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_6526_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_6527_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_6528_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_6529_sinh__0,axiom,
    aa_real_real(sinh_real,zero_zero_real) = zero_zero_real ).

% sinh_0
tff(fact_6530_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_6531_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_6532_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_6533_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_6534_inverse__less__iff__less,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_6535_inverse__less__iff__less,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_6536_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_6537_inverse__less__iff__less__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_6538_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_6539_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_6540_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_6541_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_6542_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_6543_floor__zero,axiom,
    aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_6544_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_6545_inverse__le__iff__le__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_6546_inverse__le__iff__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_6547_inverse__le__iff__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_6548_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_6549_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_6550_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_6551_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_6552_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_6553_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_6554_zero__le__floor,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_floor
tff(fact_6555_zero__le__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa) ) ).

% zero_le_floor
tff(fact_6556_floor__less__zero,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xa)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% floor_less_zero
tff(fact_6557_floor__less__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_6558_zero__less__floor,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ).

% zero_less_floor
tff(fact_6559_zero__less__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ).

% zero_less_floor
tff(fact_6560_floor__le__zero,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ).

% floor_le_zero
tff(fact_6561_floor__le__zero,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ).

% floor_le_zero
tff(fact_6562_one__le__floor,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ).

% one_le_floor
tff(fact_6563_one__le__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ).

% one_le_floor
tff(fact_6564_floor__less__one,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xa)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ).

% floor_less_one
tff(fact_6565_floor__less__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ).

% floor_less_one
tff(fact_6566_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_6567_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_6568_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_6569_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_6570_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B2) )
     => ( ( A != zero_zero_rat )
       => ( ( B2 != zero_zero_rat )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_6571_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B2) )
     => ( ( A != zero_zero_real )
       => ( ( B2 != zero_zero_real )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_6572_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_6573_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_6574_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_6575_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_6576_Ints__0,axiom,
    aa_set_real_o(member_real(zero_zero_real),ring_1_Ints_real) ).

% Ints_0
tff(fact_6577_Ints__0,axiom,
    aa_set_rat_o(member_rat(zero_zero_rat),ring_1_Ints_rat) ).

% Ints_0
tff(fact_6578_Ints__0,axiom,
    aa_set_int_o(member_int(zero_zero_int),ring_1_Ints_int) ).

% Ints_0
tff(fact_6579_floor__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim6058952711729229775r_real,Ya)) ) ).

% floor_mono
tff(fact_6580_floor__mono,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim3151403230148437115or_rat,Ya)) ) ).

% floor_mono
tff(fact_6581_of__int__floor__le,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xa))),Xa) ).

% of_int_floor_le
tff(fact_6582_of__int__floor__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xa))),Xa) ).

% of_int_floor_le
tff(fact_6583_inverse__less__imp__less,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% inverse_less_imp_less
tff(fact_6584_inverse__less__imp__less,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% inverse_less_imp_less
tff(fact_6585_less__imp__inverse__less,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_6586_less__imp__inverse__less,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_6587_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( 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,B2),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_6588_inverse__less__imp__less__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( 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,B2),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_6589_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( 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(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_6590_less__imp__inverse__less__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_6591_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_6592_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_6593_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_6594_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_6595_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_6596_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_6597_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_6598_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_6599_floor__less__cancel,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim6058952711729229775r_real,Ya))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% floor_less_cancel
tff(fact_6600_floor__less__cancel,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim3151403230148437115or_rat,Ya))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya) ) ).

% floor_less_cancel
tff(fact_6601_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_6602_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_6603_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_6604_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_6605_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_6606_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_6607_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_6608_floor__le__ceiling,axiom,
    ! [Xa: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim7802044766580827645g_real,Xa)) ).

% floor_le_ceiling
tff(fact_6609_floor__le__ceiling,axiom,
    ! [Xa: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Xa)) ).

% floor_le_ceiling
tff(fact_6610_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( aa_set_real_o(member_real(A),ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_6611_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_6612_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_6613_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_set_real_o(member_real(A),ring_1_Ints_real)
       => 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(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_6614_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_set_real_o(member_real(A),ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_rat(ring_1_of_int_rat,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_6615_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_set_real_o(member_real(A),ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2)))),ring_1_of_int_int(aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_6616_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
       => 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(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_real(ring_1_of_int_real,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_6617_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_6618_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2)))),ring_1_of_int_int(aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_6619_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( 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_eq_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_6620_le__imp__inverse__le__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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_eq_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_6621_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( 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_eq_rat,B2),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_6622_inverse__le__imp__le__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( 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_eq_real,B2),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_6623_le__imp__inverse__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_6624_le__imp__inverse__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_6625_inverse__le__imp__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% inverse_le_imp_le
tff(fact_6626_inverse__le__imp__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% inverse_le_imp_le
tff(fact_6627_inverse__le__1__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,Xa)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ) ).

% inverse_le_1_iff
tff(fact_6628_inverse__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,Xa)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% inverse_le_1_iff
tff(fact_6629_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_6630_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_6631_one__less__inverse__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xa))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_6632_one__less__inverse__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,Xa))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_6633_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_6634_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_6635_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_6636_division__ring__inverse__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))),aa_rat_rat(inverse_inverse_rat,B2)) ) ) ) ).

% division_ring_inverse_add
tff(fact_6637_division__ring__inverse__add,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))),aa_real_real(inverse_inverse_real,B2)) ) ) ) ).

% division_ring_inverse_add
tff(fact_6638_inverse__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B2)) ) ) ) ).

% inverse_add
tff(fact_6639_inverse__add,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B2)) ) ) ) ).

% inverse_add
tff(fact_6640_division__ring__inverse__diff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),A))),aa_rat_rat(inverse_inverse_rat,B2)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_6641_division__ring__inverse__diff,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))),aa_real_real(inverse_inverse_real,B2)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_6642_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_6643_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_6644_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_6645_le__floor__iff,axiom,
    ! [Z: int,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xa) ) ).

% le_floor_iff
tff(fact_6646_le__floor__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xa) ) ).

% le_floor_iff
tff(fact_6647_floor__less__iff,axiom,
    ! [Xa: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xa)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_6648_floor__less__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_6649_le__floor__add,axiom,
    ! [Xa: real,Ya: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim6058952711729229775r_real,Ya))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya))) ).

% le_floor_add
tff(fact_6650_le__floor__add,axiom,
    ! [Xa: rat,Ya: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim3151403230148437115or_rat,Ya))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya))) ).

% le_floor_add
tff(fact_6651_inverse__powr,axiom,
    ! [Ya: real,A: 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)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A)) ) ) ).

% inverse_powr
tff(fact_6652_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( aa_set_complex_o(member_complex(A),ring_1_Ints_complex)
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A)),A) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_6653_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( aa_set_real_o(member_real(A),ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_6654_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_6655_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_6656_of__nat__floor,axiom,
    ! [R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2) ) ).

% of_nat_floor
tff(fact_6657_of__nat__floor,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R2)))),R2) ) ).

% of_nat_floor
tff(fact_6658_inverse__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ) ).

% inverse_less_iff
tff(fact_6659_inverse__less__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_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,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ) ).

% inverse_less_iff
tff(fact_6660_inverse__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ) ).

% inverse_le_iff
tff(fact_6661_inverse__le__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_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,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ) ).

% inverse_le_iff
tff(fact_6662_one__le__inverse__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xa))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_6663_one__le__inverse__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,Xa))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_6664_inverse__less__1__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,Xa)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ) ).

% inverse_less_1_iff
tff(fact_6665_inverse__less__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,Xa)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% inverse_less_1_iff
tff(fact_6666_one__le__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_le_inverse
tff(fact_6667_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_6668_inverse__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2))),aa_rat_rat(inverse_inverse_rat,B2))) ) ) ) ).

% inverse_diff_inverse
tff(fact_6669_inverse__diff__inverse,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B2))),aa_complex_complex(invers8013647133539491842omplex,B2))) ) ) ) ).

% inverse_diff_inverse
tff(fact_6670_inverse__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))),aa_real_real(inverse_inverse_real,B2))) ) ) ) ).

% inverse_diff_inverse
tff(fact_6671_reals__Archimedean,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),Xa) ) ).

% reals_Archimedean
tff(fact_6672_reals__Archimedean,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),Xa) ) ).

% reals_Archimedean
tff(fact_6673_le__mult__nat__floor,axiom,
    ! [A: real,B2: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ).

% le_mult_nat_floor
tff(fact_6674_le__mult__nat__floor,axiom,
    ! [A: rat,B2: rat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,A))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ).

% le_mult_nat_floor
tff(fact_6675_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_6676_floor__eq3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Na ) ) ) ).

% floor_eq3
tff(fact_6677_le__nat__floor,axiom,
    ! [Xa: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_6678_ceiling__diff__floor__le__1,axiom,
    ! [Xa: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_real_int(archim6058952711729229775r_real,Xa))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_6679_ceiling__diff__floor__le__1,axiom,
    ! [Xa: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_rat_int(archim3151403230148437115or_rat,Xa))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_6680_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_6681_floor__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Na ) ) ) ).

% floor_eq
tff(fact_6682_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_6683_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_6684_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_6685_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D5: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D5),E)
         => ( aa_real_o(P,D5)
           => aa_real_o(P,E) ) )
     => ( ! [N2: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_6686_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N: nat] :
          ( ( N != 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,N)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))),E2) ) ) ).

% real_arch_inverse
tff(fact_6687_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D5: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D5),E)
         => ( aa_real_o(P,D5)
           => aa_real_o(P,E) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_6688_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_inverse
tff(fact_6689_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( aa_set_real_o(member_real(A),ring_1_Ints_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(plus_plus_real,one_one_real),A)),A)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% Ints_odd_less_0
tff(fact_6690_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_6691_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_6692_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: code_integer] :
      ( member_Code_integer(Xa,ring_11222124179247155820nteger)
     => ( ( Xa != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6693_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: real] :
      ( aa_set_real_o(member_real(Xa),ring_1_Ints_real)
     => ( ( Xa != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6694_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: rat] :
      ( aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat)
     => ( ( Xa != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6695_Ints__nonzero__abs__ge1,axiom,
    ! [Xa: int] :
      ( aa_set_int_o(member_int(Xa),ring_1_Ints_int)
     => ( ( Xa != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,Xa)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6696_Ints__nonzero__abs__less1,axiom,
    ! [Xa: code_integer] :
      ( member_Code_integer(Xa,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xa)),one_one_Code_integer)
       => ( Xa = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6697_Ints__nonzero__abs__less1,axiom,
    ! [Xa: real] :
      ( aa_set_real_o(member_real(Xa),ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
       => ( Xa = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6698_Ints__nonzero__abs__less1,axiom,
    ! [Xa: rat] :
      ( aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xa)),one_one_rat)
       => ( Xa = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6699_Ints__nonzero__abs__less1,axiom,
    ! [Xa: int] :
      ( aa_set_int_o(member_int(Xa),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xa)),one_one_int)
       => ( Xa = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6700_Ints__eq__abs__less1,axiom,
    ! [Xa: code_integer,Ya: code_integer] :
      ( member_Code_integer(Xa,ring_11222124179247155820nteger)
     => ( member_Code_integer(Ya,ring_11222124179247155820nteger)
       => ( ( Xa = Ya )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,Ya))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6701_Ints__eq__abs__less1,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_set_real_o(member_real(Xa),ring_1_Ints_real)
     => ( aa_set_real_o(member_real(Ya),ring_1_Ints_real)
       => ( ( Xa = Ya )
        <=> 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,Xa),Ya))),one_one_real) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6702_Ints__eq__abs__less1,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat)
     => ( aa_set_rat_o(member_rat(Ya),ring_1_Ints_rat)
       => ( ( Xa = Ya )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),Ya))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6703_Ints__eq__abs__less1,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_set_int_o(member_int(Xa),ring_1_Ints_int)
     => ( aa_set_int_o(member_int(Ya),ring_1_Ints_int)
       => ( ( Xa = Ya )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Ya))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6704_floor__unique,axiom,
    ! [Z: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Z ) ) ) ).

% floor_unique
tff(fact_6705_floor__unique,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,Xa) = Z ) ) ) ).

% floor_unique
tff(fact_6706_floor__eq__iff,axiom,
    ! [Xa: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,Xa) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_6707_floor__eq__iff,axiom,
    ! [Xa: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,Xa) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_6708_floor__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Ta))
    <=> ! [I4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)) )
         => aa_int_o(P,I4) ) ) ).

% floor_split
tff(fact_6709_floor__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim3151403230148437115or_rat,Ta))
    <=> ! [I4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)) )
         => aa_int_o(P,I4) ) ) ).

% floor_split
tff(fact_6710_le__mult__floor,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2))) ) ) ).

% le_mult_floor
tff(fact_6711_le__mult__floor,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))) ) ) ).

% le_mult_floor
tff(fact_6712_less__floor__iff,axiom,
    ! [Z: int,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> 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,Z)),one_one_real)),Xa) ) ).

% less_floor_iff
tff(fact_6713_less__floor__iff,axiom,
    ! [Z: int,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xa) ) ).

% less_floor_iff
tff(fact_6714_floor__le__iff,axiom,
    ! [Xa: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_6715_floor__le__iff,axiom,
    ! [Xa: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_6716_floor__correct,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xa))),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xa)),one_one_int))) ) ).

% floor_correct
tff(fact_6717_floor__correct,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xa))),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),one_one_int))) ) ).

% floor_correct
tff(fact_6718_ex__inverse__of__nat__less,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),Xa) ) ) ).

% ex_inverse_of_nat_less
tff(fact_6719_ex__inverse__of__nat__less,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),Xa) ) ) ).

% ex_inverse_of_nat_less
tff(fact_6720_power__diff__conv__inverse,axiom,
    ! [Xa: complex,M2: nat,Na: nat] :
      ( ( Xa != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_nat_complex(power_power_complex(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xa),Na)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xa)),M2)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_6721_power__diff__conv__inverse,axiom,
    ! [Xa: rat,M2: nat,Na: nat] :
      ( ( Xa != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_nat_rat(power_power_rat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xa),Na)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xa)),M2)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_6722_power__diff__conv__inverse,axiom,
    ! [Xa: real,M2: nat,Na: nat] :
      ( ( Xa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xa)),M2)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_6723_floor__eq4,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Na ) ) ) ).

% floor_eq4
tff(fact_6724_floor__eq2,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Na ) ) ) ).

% floor_eq2
tff(fact_6725_floor__divide__real__eq__div,axiom,
    ! [B2: int,A: 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,A),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,A)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_6726_log__inverse,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_6727_floor__divide__lower,axiom,
    ! [Q5: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q5)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q5)))),Q5)),P3) ) ).

% floor_divide_lower
tff(fact_6728_floor__divide__lower,axiom,
    ! [Q5: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q5)))),Q5)),P3) ) ).

% floor_divide_lower
tff(fact_6729_Cauchy__iff2,axiom,
    ! [X3: fun_nat_real] :
      ( topolo4055970368930404560y_real(X3)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M)
         => ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N)
             => 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(X3,M)),aa_nat_real(X3,N)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_6730_floor__add,axiom,
    ! [Xa: real,Ya: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Ya))),one_one_real),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim6058952711729229775r_real,Ya)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim6058952711729229775r_real,Ya))),one_one_int)) ).

% floor_add
tff(fact_6731_floor__add,axiom,
    ! [Xa: rat,Ya: rat] :
      aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Ya))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim3151403230148437115or_rat,Ya)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim3151403230148437115or_rat,Ya))),one_one_int)) ).

% floor_add
tff(fact_6732_frac__unique__iff,axiom,
    ! [Xa: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xa) = A )
    <=> ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),A)),ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) ) ) ).

% frac_unique_iff
tff(fact_6733_frac__unique__iff,axiom,
    ! [Xa: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xa) = A )
    <=> ( aa_set_rat_o(member_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),A)),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_6734_frac__neg,axiom,
    ! [Xa: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,Xa)) = $ite(aa_set_real_o(member_real(Xa),ring_1_Ints_real),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(archim2898591450579166408c_real,Xa))) ).

% frac_neg
tff(fact_6735_frac__neg,axiom,
    ! [Xa: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,Xa)) = $ite(aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(archimedean_frac_rat,Xa))) ).

% frac_neg
tff(fact_6736_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_6737_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_6738_verit__le__mono__div__int,axiom,
    ! [A2: int,B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_6739_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_6740_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_6741_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_6742_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_6743_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_6744_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_6745_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_6746_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_6747_mod__mult__self1__is__0,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),B2) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_6748_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),B2) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_6749_mod__mult__self2__is__0,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),B2) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_6750_mod__mult__self2__is__0,axiom,
    ! [A: 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,A),B2)),B2) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_6751_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_6752_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_6753_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_6754_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_6755_mod__div__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) = zero_zero_int ).

% mod_div_trivial
tff(fact_6756_mod__div__trivial,axiom,
    ! [A: 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,A),B2)),B2) = zero_zero_nat ).

% mod_div_trivial
tff(fact_6757_bits__mod__div__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_6758_bits__mod__div__trivial,axiom,
    ! [A: 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,A),B2)),B2) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_6759_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_6760_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_6761_frac__eq__0__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xa) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xa),ring_1_Ints_real) ) ).

% frac_eq_0_iff
tff(fact_6762_frac__eq__0__iff,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xa) = zero_zero_rat )
    <=> aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_6763_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),uminus_uminus_int(one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_6764_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_6765_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_6766_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_6767_frac__gt__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xa))
    <=> ~ aa_set_real_o(member_real(Xa),ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_6768_frac__gt__0__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xa))
    <=> ~ aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_6769_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6770_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6771_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A: 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(modulo_modulo_int,A),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6772_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A: 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,A),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6773_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6774_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6775_zmod__le__nonneg__dividend,axiom,
    ! [M2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M2),K)),M2) ) ).

% zmod_le_nonneg_dividend
tff(fact_6776_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_6777_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_6778_frac__ge__0,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xa)) ).

% frac_ge_0
tff(fact_6779_frac__ge__0,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xa)) ).

% frac_ge_0
tff(fact_6780_frac__lt__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xa)),one_one_real) ).

% frac_lt_1
tff(fact_6781_frac__lt__1,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,Xa)),one_one_rat) ).

% frac_lt_1
tff(fact_6782_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6783_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6784_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A: 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,A),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6785_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A: 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(modulo_modulo_int,A),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6786_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_6787_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_6788_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_6789_pos__mod__conj,axiom,
    ! [B2: int,A: 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(modulo_modulo_int,A),B2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_6790_neg__mod__conj,axiom,
    ! [B2: int,A: 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(aa_int_fun_int_int(modulo_modulo_int,A),B2)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ) ).

% neg_mod_conj
tff(fact_6791_zdiv__mono__strict,axiom,
    ! [A2: int,B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Na) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_6792_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_6793_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_6794_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_6795_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: 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,A),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,A),B2)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6796_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: 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(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6797_frac__eq,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xa) = Xa )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% frac_eq
tff(fact_6798_frac__eq,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xa) = Xa )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ) ).

% frac_eq
tff(fact_6799_int__mod__pos__eq,axiom,
    ! [A: int,B2: int,Q5: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_6800_int__mod__neg__eq,axiom,
    ! [A: int,B2: int,Q5: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_6801_split__zmod,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_6802_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,uminus_uminus_int(K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_6803_frac__add,axiom,
    ! [Xa: real,Ya: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Ya))),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Ya)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Ya))),one_one_real)) ).

% frac_add
tff(fact_6804_frac__add,axiom,
    ! [Xa: rat,Ya: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Ya))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Ya)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Ya))),one_one_rat)) ).

% frac_add
tff(fact_6805_zmod__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(modulo_modulo_int,uminus_uminus_int(one_one_int)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),one_one_int) ) ) ).

% zmod_minus1
tff(fact_6806_zmod__zmult2__eq,axiom,
    ! [C2: int,A: 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(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_6807_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6808_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6809_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6810_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(semiri2265585572941072030t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6811_fact__num__eq__if,axiom,
    ! [M2: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,M2) = $ite(M2 = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6812_fact__num__eq__if,axiom,
    ! [M2: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,M2) = $ite(M2 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6813_fact__num__eq__if,axiom,
    ! [M2: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,M2) = $ite(M2 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6814_fact__num__eq__if,axiom,
    ! [M2: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,M2) = $ite(M2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6815_fact__num__eq__if,axiom,
    ! [M2: nat] :
      aa_nat_real(semiri2265585572941072030t_real,M2) = $ite(M2 = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6816_verit__le__mono__div,axiom,
    ! [A2: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Na)) ) ) ).

% verit_le_mono_div
tff(fact_6817_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W2: 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,W2)))
    <=> $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,W2))),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,W2))),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,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6818_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6819_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),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,W2))),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,W2))),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,W2))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6820_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6821_norm__power__diff,axiom,
    ! [Z: real,W2: real,M2: 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,W2)),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),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),M2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W2)))) ) ) ).

% norm_power_diff
tff(fact_6822_norm__power__diff,axiom,
    ! [Z: complex,W2: complex,M2: 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,W2)),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),M2)),aa_nat_complex(power_power_complex(W2),M2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W2)))) ) ) ).

% norm_power_diff
tff(fact_6823_Gcd__fin__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( semiri4258706085729940814in_nat(A2) = zero_zero_nat )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(zero_zero_nat,bot_bot_set_nat))
        & aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% Gcd_fin_0_iff
tff(fact_6824_Gcd__fin__0__iff,axiom,
    ! [A2: set_int] :
      ( ( semiri4256215615220890538in_int(A2) = zero_zero_int )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(zero_zero_int,bot_bot_set_int))
        & aa_set_int_o(finite_finite_int,A2) ) ) ).

% Gcd_fin_0_iff
tff(fact_6825_numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_num_real(numeral_numeral_real,M2) = aa_num_real(numeral_numeral_real,Na) )
    <=> ( M2 = Na ) ) ).

% numeral_eq_iff
tff(fact_6826_numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M2) = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( M2 = Na ) ) ).

% numeral_eq_iff
tff(fact_6827_numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_num_int(numeral_numeral_int,M2) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( M2 = Na ) ) ).

% numeral_eq_iff
tff(fact_6828_numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( numera1916890842035813515d_enat(M2) = numera1916890842035813515d_enat(Na) )
    <=> ( M2 = Na ) ) ).

% numeral_eq_iff
tff(fact_6829_numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( numera6620942414471956472nteger(M2) = numera6620942414471956472nteger(Na) )
    <=> ( M2 = Na ) ) ).

% numeral_eq_iff
tff(fact_6830_numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% numeral_le_iff
tff(fact_6831_numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% numeral_le_iff
tff(fact_6832_numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% numeral_le_iff
tff(fact_6833_numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% numeral_le_iff
tff(fact_6834_numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% numeral_le_iff
tff(fact_6835_numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% numeral_le_iff
tff(fact_6836_numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% numeral_less_iff
tff(fact_6837_numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% numeral_less_iff
tff(fact_6838_numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% numeral_less_iff
tff(fact_6839_numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% numeral_less_iff
tff(fact_6840_numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% numeral_less_iff
tff(fact_6841_numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% numeral_less_iff
tff(fact_6842_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_6843_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_6844_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),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),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_6845_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W2)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_6846_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),times_7803423173614009249d_enat(numera1916890842035813515d_enat(W2),Z)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(times_times_num(V),W2)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_6847_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),times_3573771949741848930nteger(numera6620942414471956472nteger(W2),Z)) = times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(V),W2)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_6848_numeral__times__numeral,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M2),Na)) ).

% numeral_times_numeral
tff(fact_6849_numeral__times__numeral,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M2),Na)) ).

% numeral_times_numeral
tff(fact_6850_numeral__times__numeral,axiom,
    ! [M2: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M2),Na)) ).

% numeral_times_numeral
tff(fact_6851_numeral__times__numeral,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M2),Na)) ).

% numeral_times_numeral
tff(fact_6852_numeral__times__numeral,axiom,
    ! [M2: num,Na: num] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(times_times_num(M2),Na)) ).

% numeral_times_numeral
tff(fact_6853_numeral__times__numeral,axiom,
    ! [M2: num,Na: num] : times_3573771949741848930nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(Na)) = numera6620942414471956472nteger(aa_num_num(times_times_num(M2),Na)) ).

% numeral_times_numeral
tff(fact_6854_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_6855_add__numeral__left,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_6856_add__numeral__left,axiom,
    ! [V: num,W2: 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,W2)),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),W2))),Z) ).

% add_numeral_left
tff(fact_6857_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W2)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_6858_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2),Z)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(plus_plus_num(V),W2)),Z) ).

% add_numeral_left
tff(fact_6859_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: code_integer] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(V),plus_p5714425477246183910nteger(numera6620942414471956472nteger(W2),Z)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W2)),Z) ).

% add_numeral_left
tff(fact_6860_numeral__plus__numeral,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M2),Na)) ).

% numeral_plus_numeral
tff(fact_6861_numeral__plus__numeral,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M2),Na)) ).

% numeral_plus_numeral
tff(fact_6862_numeral__plus__numeral,axiom,
    ! [M2: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M2),Na)) ).

% numeral_plus_numeral
tff(fact_6863_numeral__plus__numeral,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M2),Na)) ).

% numeral_plus_numeral
tff(fact_6864_numeral__plus__numeral,axiom,
    ! [M2: num,Na: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(M2),Na)) ).

% numeral_plus_numeral
tff(fact_6865_numeral__plus__numeral,axiom,
    ! [M2: num,Na: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(Na)) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(M2),Na)) ).

% numeral_plus_numeral
tff(fact_6866_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_6867_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_6868_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_6869_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_6870_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_6871_neg__numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)) )
    <=> ( M2 = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6872_neg__numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) )
    <=> ( M2 = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6873_neg__numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) )
    <=> ( M2 = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6874_neg__numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) )
    <=> ( M2 = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6875_neg__numeral__eq__iff,axiom,
    ! [M2: num,Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) )
    <=> ( M2 = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_6876_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,Na) ).

% of_nat_numeral
tff(fact_6877_of__nat__numeral,axiom,
    ! [Na: num] : semiri4216267220026989637d_enat(aa_num_nat(numeral_numeral_nat,Na)) = numera1916890842035813515d_enat(Na) ).

% of_nat_numeral
tff(fact_6878_of__nat__numeral,axiom,
    ! [Na: num] : semiri4939895301339042750nteger(aa_num_nat(numeral_numeral_nat,Na)) = numera6620942414471956472nteger(Na) ).

% of_nat_numeral
tff(fact_6879_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% of_nat_numeral
tff(fact_6880_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_real(numeral_numeral_real,Na) ).

% of_nat_numeral
tff(fact_6881_abs__numeral,axiom,
    ! [Na: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,Na) ).

% abs_numeral
tff(fact_6882_abs__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,Na) ).

% abs_numeral
tff(fact_6883_abs__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% abs_numeral
tff(fact_6884_abs__numeral,axiom,
    ! [Na: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Na)) = numera6620942414471956472nteger(Na) ).

% abs_numeral
tff(fact_6885_mod__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na) = M2 ) ) ).

% mod_less
tff(fact_6886_neg__numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% neg_numeral_le_iff
tff(fact_6887_neg__numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% neg_numeral_le_iff
tff(fact_6888_neg__numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% neg_numeral_le_iff
tff(fact_6889_neg__numeral__le__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% neg_numeral_le_iff
tff(fact_6890_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% distrib_left_numeral
tff(fact_6891_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_6892_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_6893_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% distrib_left_numeral
tff(fact_6894_distrib__left__numeral,axiom,
    ! [V: num,B2: extended_enat,C2: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(B2,C2)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),B2),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),C2)) ).

% distrib_left_numeral
tff(fact_6895_distrib__left__numeral,axiom,
    ! [V: num,B2: code_integer,C2: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),plus_p5714425477246183910nteger(B2,C2)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V),B2),times_3573771949741848930nteger(numera6620942414471956472nteger(V),C2)) ).

% distrib_left_numeral
tff(fact_6896_distrib__right__numeral,axiom,
    ! [A: rat,B2: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_6897_distrib__right__numeral,axiom,
    ! [A: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),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,A),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_6898_distrib__right__numeral,axiom,
    ! [A: 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,A),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,A),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_6899_distrib__right__numeral,axiom,
    ! [A: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_6900_distrib__right__numeral,axiom,
    ! [A: extended_enat,B2: extended_enat,V: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(A,B2),numera1916890842035813515d_enat(V)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(V)),times_7803423173614009249d_enat(B2,numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_6901_distrib__right__numeral,axiom,
    ! [A: code_integer,B2: code_integer,V: num] : times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B2),numera6620942414471956472nteger(V)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(A,numera6620942414471956472nteger(V)),times_3573771949741848930nteger(B2,numera6620942414471956472nteger(V))) ).

% distrib_right_numeral
tff(fact_6902_neg__numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% neg_numeral_less_iff
tff(fact_6903_neg__numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% neg_numeral_less_iff
tff(fact_6904_neg__numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% neg_numeral_less_iff
tff(fact_6905_neg__numeral__less__iff,axiom,
    ! [M2: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% neg_numeral_less_iff
tff(fact_6906_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B2),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_6907_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_6908_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_6909_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: code_integer,C2: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),minus_8373710615458151222nteger(B2,C2)) = minus_8373710615458151222nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V),B2),times_3573771949741848930nteger(numera6620942414471956472nteger(V),C2)) ).

% right_diff_distrib_numeral
tff(fact_6910_left__diff__distrib__numeral,axiom,
    ! [A: rat,B2: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_6911_left__diff__distrib__numeral,axiom,
    ! [A: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),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,A),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_6912_left__diff__distrib__numeral,axiom,
    ! [A: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_6913_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B2: code_integer,V: num] : times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B2),numera6620942414471956472nteger(V)) = minus_8373710615458151222nteger(times_3573771949741848930nteger(A,numera6620942414471956472nteger(V)),times_3573771949741848930nteger(B2,numera6620942414471956472nteger(V))) ).

% left_diff_distrib_numeral
tff(fact_6914_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6915_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6916_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6917_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : times_3573771949741848930nteger(numera6620942414471956472nteger(M2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6918_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M2)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_6919_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,Na)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6920_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6921_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6922_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6923_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M2),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_6924_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M2),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6925_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M2),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6926_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M2),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6927_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(aa_num_num(times_times_num(M2),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6928_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(times_times_num(M2),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_6929_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = uminus_uminus_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6930_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6931_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6932_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6933_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M2)),numera6690914467698888265omplex(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_6934_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,Na)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M2),Na))) ).

% diff_numeral_simps(3)
tff(fact_6935_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M2),Na))) ).

% diff_numeral_simps(3)
tff(fact_6936_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M2),Na))) ).

% diff_numeral_simps(3)
tff(fact_6937_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(M2),Na))) ).

% diff_numeral_simps(3)
tff(fact_6938_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M2),Na))) ).

% diff_numeral_simps(3)
tff(fact_6939_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M2),Na)) ).

% diff_numeral_simps(2)
tff(fact_6940_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M2),Na)) ).

% diff_numeral_simps(2)
tff(fact_6941_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M2),Na)) ).

% diff_numeral_simps(2)
tff(fact_6942_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(M2),Na)) ).

% diff_numeral_simps(2)
tff(fact_6943_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M2)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M2),Na)) ).

% diff_numeral_simps(2)
tff(fact_6944_abs__neg__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,Na) ).

% abs_neg_numeral
tff(fact_6945_abs__neg__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,Na) ).

% abs_neg_numeral
tff(fact_6946_abs__neg__numeral,axiom,
    ! [Na: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,Na) ).

% abs_neg_numeral
tff(fact_6947_abs__neg__numeral,axiom,
    ! [Na: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(Na) ).

% abs_neg_numeral
tff(fact_6948_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_6949_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_6950_norm__eq__zero,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_6951_norm__eq__zero,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_6952_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_6953_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_6954_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_6955_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_6956_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_6957_mod__by__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_6958_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_6959_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_num_real(numeral_numeral_real,W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_6960_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_nat_real(semiri5074537144036343181t_real,M2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),M2) ) ).

% numeral_le_real_of_nat_iff
tff(fact_6961_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,uminus_uminus_int(aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_6962_Gcd__fin_Oempty,axiom,
    semiri4258706085729940814in_nat(bot_bot_set_nat) = zero_zero_nat ).

% Gcd_fin.empty
tff(fact_6963_Gcd__fin_Oempty,axiom,
    semiri4256215615220890538in_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_fin.empty
tff(fact_6964_Gcd__fin_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( semiri4258706085729940814in_nat(A2) = one_one_nat ) ) ).

% Gcd_fin.infinite
tff(fact_6965_Gcd__fin_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( semiri4256215615220890538in_int(A2) = one_one_int ) ) ).

% Gcd_fin.infinite
tff(fact_6966_Gcd__fin__eq__Gcd,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( semiri4256215615220890538in_int(A2) = gcd_Gcd_int(A2) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_6967_Gcd__fin__eq__Gcd,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( semiri4258706085729940814in_nat(A2) = gcd_Gcd_nat(A2) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_6968_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_6969_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_6970_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: 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,W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_6971_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_num_rat(numeral_numeral_rat,W2))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_6972_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W2: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)) = B2,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_6973_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) = B2,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_6974_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B2: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W2)) )
    <=> $ite(numera6690914467698888265omplex(W2) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B2,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_6975_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_6976_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W2)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_6977_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W2)) = A )
    <=> $ite(numera6690914467698888265omplex(W2) != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_6978_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_6979_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_6980_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_num_rat(numeral_numeral_rat,W2))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_6981_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: 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,W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_6982_inverse__eq__divide__numeral,axiom,
    ! [W2: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,W2)) ).

% inverse_eq_divide_numeral
tff(fact_6983_inverse__eq__divide__numeral,axiom,
    ! [W2: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W2)) ).

% inverse_eq_divide_numeral
tff(fact_6984_inverse__eq__divide__numeral,axiom,
    ! [W2: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,W2)) ).

% inverse_eq_divide_numeral
tff(fact_6985_zero__less__norm__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xa))
    <=> ( Xa != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6986_zero__less__norm__iff,axiom,
    ! [Xa: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xa))
    <=> ( Xa != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6987_norm__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xa)),zero_zero_real)
    <=> ( Xa = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6988_norm__le__zero__iff,axiom,
    ! [Xa: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),zero_zero_real)
    <=> ( Xa = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6989_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6990_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(Z)),numera6620942414471956472nteger(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6991_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6992_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6993_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6994_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Na)),ring_18347121197199848620nteger(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6995_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6996_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6997_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6998_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6999_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_7000_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(Z)),numera6620942414471956472nteger(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_7001_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7002_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7003_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),ring_1_of_int_int(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7004_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Na)),ring_18347121197199848620nteger(Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7005_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7006_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7007_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7008_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7009_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7010_numeral__le__floor,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),Xa) ) ).

% numeral_le_floor
tff(fact_7011_numeral__le__floor,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),Xa) ) ).

% numeral_le_floor
tff(fact_7012_floor__less__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,V)) ) ).

% floor_less_numeral
tff(fact_7013_floor__less__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_num_rat(numeral_numeral_rat,V)) ) ).

% floor_less_numeral
tff(fact_7014_ceiling__le__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_7015_ceiling__le__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_7016_numeral__less__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xa) ) ).

% numeral_less_ceiling
tff(fact_7017_numeral__less__ceiling,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),Xa) ) ).

% numeral_less_ceiling
tff(fact_7018_powr__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_num_real(numeral_numeral_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Na)) ) ) ).

% powr_numeral
tff(fact_7019_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: 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,W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_7020_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_7021_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_7022_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_7023_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_7024_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_7025_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_7026_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B2,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_7027_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W2: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B2,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_7028_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B2: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B2,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_7029_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: 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,W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_7030_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_7031_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_7032_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_7033_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu3811975205180677377ec_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,K))) = uminus_uminus_int(neg_nu5851722552734809277nc_int(aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_7034_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_7035_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_7036_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).

% dbl_dec_simps(1)
tff(fact_7037_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_7038_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5851722552734809277nc_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,K))) = uminus_uminus_int(neg_nu3811975205180677377ec_int(aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_7039_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_7040_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_7041_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).

% dbl_inc_simps(1)
tff(fact_7042_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_7043_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7044_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7045_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7046_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_7047_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_7048_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_7049_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_7050_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7051_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7052_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,semiri4939895301339042750nteger(Xa)),power_8256067586552552935nteger(numera6620942414471956472nteger(I),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7053_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7054_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7055_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7056_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7057_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(I),Na)),semiri4939895301339042750nteger(Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7058_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7059_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7060_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,semiri4939895301339042750nteger(Xa)),power_8256067586552552935nteger(numera6620942414471956472nteger(I),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7061_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7062_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7063_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7064_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xa: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7065_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(I),Na)),semiri4939895301339042750nteger(Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7066_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7067_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7068_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7069_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),Xa) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7070_numeral__less__floor,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xa) ) ).

% numeral_less_floor
tff(fact_7071_numeral__less__floor,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xa) ) ).

% numeral_less_floor
tff(fact_7072_floor__le__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_7073_floor__le__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_7074_ceiling__less__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_7075_ceiling__less__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_7076_numeral__le__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xa) ) ).

% numeral_le_ceiling
tff(fact_7077_numeral__le__ceiling,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xa) ) ).

% numeral_le_ceiling
tff(fact_7078_neg__numeral__le__floor,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xa) ) ).

% neg_numeral_le_floor
tff(fact_7079_neg__numeral__le__floor,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xa) ) ).

% neg_numeral_le_floor
tff(fact_7080_floor__less__neg__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_neg_numeral
tff(fact_7081_floor__less__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_neg_numeral
tff(fact_7082_ceiling__le__neg__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_7083_ceiling__le__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_7084_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7085_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A)),power_8256067586552552935nteger(numera6620942414471956472nteger(Xa),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7086_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7087_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7088_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7089_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(Xa),Na)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7090_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7091_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7092_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xa) ) ).

% neg_numeral_less_ceiling
tff(fact_7093_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xa) ) ).

% neg_numeral_less_ceiling
tff(fact_7094_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7095_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7096_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7097_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(Xa),Na)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7098_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7099_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7100_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7101_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A)),power_8256067586552552935nteger(numera6620942414471956472nteger(Xa),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7102_neg__numeral__less__floor,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xa) ) ).

% neg_numeral_less_floor
tff(fact_7103_neg__numeral__less__floor,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xa) ) ).

% neg_numeral_less_floor
tff(fact_7104_floor__le__neg__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_7105_floor__le__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_7106_ceiling__less__neg__numeral,axiom,
    ! [Xa: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(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_7107_ceiling__less__neg__numeral,axiom,
    ! [Xa: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xa)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_7108_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xa) ) ).

% neg_numeral_le_ceiling
tff(fact_7109_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xa) ) ).

% neg_numeral_le_ceiling
tff(fact_7110_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7111_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7112_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7113_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7114_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7115_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa)),Na)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7116_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7117_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7118_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),ring_1_of_int_int(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7119_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7120_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7121_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa)),Na)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7122_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,ring_1_of_int_int(A)),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7123_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7124_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7125_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,Xa))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7126_fact__ge__self,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_self
tff(fact_7127_fact__mono__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono_nat
tff(fact_7128_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) != zero_zero_rat ).

% fact_nonzero
tff(fact_7129_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) != zero_zero_int ).

% fact_nonzero
tff(fact_7130_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) != zero_zero_nat ).

% fact_nonzero
tff(fact_7131_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) != zero_zero_real ).

% fact_nonzero
tff(fact_7132_mod__less__eq__dividend,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na)),M2) ).

% mod_less_eq_dividend
tff(fact_7133_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Na) ).

% zero_neq_numeral
tff(fact_7134_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Na) ).

% zero_neq_numeral
tff(fact_7135_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Na) ).

% zero_neq_numeral
tff(fact_7136_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Na) ).

% zero_neq_numeral
tff(fact_7137_zero__neq__numeral,axiom,
    ! [Na: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(Na) ).

% zero_neq_numeral
tff(fact_7138_zero__neq__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != numera6620942414471956472nteger(Na) ).

% zero_neq_numeral
tff(fact_7139_numeral__neq__neg__numeral,axiom,
    ! [M2: num,Na: num] : aa_num_int(numeral_numeral_int,M2) != uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)) ).

% numeral_neq_neg_numeral
tff(fact_7140_numeral__neq__neg__numeral,axiom,
    ! [M2: num,Na: num] : aa_num_real(numeral_numeral_real,M2) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% numeral_neq_neg_numeral
tff(fact_7141_numeral__neq__neg__numeral,axiom,
    ! [M2: num,Na: num] : aa_num_rat(numeral_numeral_rat,M2) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_neq_neg_numeral
tff(fact_7142_numeral__neq__neg__numeral,axiom,
    ! [M2: num,Na: num] : numera6620942414471956472nteger(M2) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% numeral_neq_neg_numeral
tff(fact_7143_numeral__neq__neg__numeral,axiom,
    ! [M2: num,Na: num] : numera6690914467698888265omplex(M2) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% numeral_neq_neg_numeral
tff(fact_7144_neg__numeral__neq__numeral,axiom,
    ! [M2: num,Na: num] : uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)) != aa_num_int(numeral_numeral_int,Na) ).

% neg_numeral_neq_numeral
tff(fact_7145_neg__numeral__neq__numeral,axiom,
    ! [M2: num,Na: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)) != aa_num_real(numeral_numeral_real,Na) ).

% neg_numeral_neq_numeral
tff(fact_7146_neg__numeral__neq__numeral,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)) != aa_num_rat(numeral_numeral_rat,Na) ).

% neg_numeral_neq_numeral
tff(fact_7147_neg__numeral__neq__numeral,axiom,
    ! [M2: num,Na: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)) != numera6620942414471956472nteger(Na) ).

% neg_numeral_neq_numeral
tff(fact_7148_neg__numeral__neq__numeral,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2)) != numera6690914467698888265omplex(Na) ).

% neg_numeral_neq_numeral
tff(fact_7149_fact__less__mono__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_7150_fact__ge__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_zero
tff(fact_7151_fact__ge__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_zero
tff(fact_7152_fact__ge__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_zero
tff(fact_7153_fact__ge__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_zero
tff(fact_7154_fact__gt__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_gt_zero
tff(fact_7155_fact__gt__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_gt_zero
tff(fact_7156_fact__gt__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_gt_zero
tff(fact_7157_fact__gt__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_gt_zero
tff(fact_7158_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),zero_zero_rat) ).

% fact_not_neg
tff(fact_7159_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),zero_zero_int) ).

% fact_not_neg
tff(fact_7160_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),zero_zero_nat) ).

% fact_not_neg
tff(fact_7161_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Na)),zero_zero_real) ).

% fact_not_neg
tff(fact_7162_norm__not__less__zero,axiom,
    ! [Xa: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_7163_mod__Suc,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),Na) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na)) = Na,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na))) ).

% mod_Suc
tff(fact_7164_norm__ge__zero,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xa)) ).

% norm_ge_zero
tff(fact_7165_fact__ge__1,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_1
tff(fact_7166_fact__ge__1,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_1
tff(fact_7167_fact__ge__1,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_1
tff(fact_7168_fact__ge__1,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_1
tff(fact_7169_mod__induct,axiom,
    ! [P: fun_nat_o,Na: nat,P3: nat,M2: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),P3)
         => ( ! [N2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),P3)
               => ( aa_nat_o(P,N2)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3)) ) )
           => aa_nat_o(P,M2) ) ) ) ) ).

% mod_induct
tff(fact_7170_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M2: nat,Na: nat] :
      ( ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M4),zero_zero_nat)
     => ( ! [M4: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M4),N2))
             => aa_nat_o(aa_nat_fun_nat_o(P,M4),N2) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M2),Na) ) ) ).

% gcd_nat_induct
tff(fact_7171_mod__less__divisor,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na)),Na) ) ).

% mod_less_divisor
tff(fact_7172_mod__Suc__le__divisor,axiom,
    ! [M2: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_7173_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_7174_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Na)),zero_z5237406670263579293d_enat) ).

% not_numeral_le_zero
tff(fact_7175_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Na)),zero_z3403309356797280102nteger) ).

% not_numeral_le_zero
tff(fact_7176_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_7177_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_7178_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_7179_zero__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_le_numeral
tff(fact_7180_zero__le__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Na)) ).

% zero_le_numeral
tff(fact_7181_zero__le__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(Na)) ).

% zero_le_numeral
tff(fact_7182_zero__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_le_numeral
tff(fact_7183_zero__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_le_numeral
tff(fact_7184_zero__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_le_numeral
tff(fact_7185_zero__less__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_less_numeral
tff(fact_7186_zero__less__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_less_numeral
tff(fact_7187_zero__less__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_less_numeral
tff(fact_7188_zero__less__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_less_numeral
tff(fact_7189_zero__less__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Na)) ).

% zero_less_numeral
tff(fact_7190_zero__less__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(Na)) ).

% zero_less_numeral
tff(fact_7191_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_7192_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_7193_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_7194_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_7195_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Na)),zero_z5237406670263579293d_enat) ).

% not_numeral_less_zero
tff(fact_7196_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Na)),zero_z3403309356797280102nteger) ).

% not_numeral_less_zero
tff(fact_7197_fact__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% fact_mono
tff(fact_7198_fact__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% fact_mono
tff(fact_7199_fact__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono
tff(fact_7200_fact__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M2)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% fact_mono
tff(fact_7201_one__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) ).

% one_le_numeral
tff(fact_7202_one__le__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Na)) ).

% one_le_numeral
tff(fact_7203_one__le__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),numera6620942414471956472nteger(Na)) ).

% one_le_numeral
tff(fact_7204_one__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% one_le_numeral
tff(fact_7205_one__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% one_le_numeral
tff(fact_7206_one__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) ).

% one_le_numeral
tff(fact_7207_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),D) = zero_zero_nat )
     => ? [Q3: nat] : M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_7208_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) ).

% not_numeral_less_one
tff(fact_7209_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) ).

% not_numeral_less_one
tff(fact_7210_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) ).

% not_numeral_less_one
tff(fact_7211_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) ).

% not_numeral_less_one
tff(fact_7212_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat) ).

% not_numeral_less_one
tff(fact_7213_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Na)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_7214_mod__geq,axiom,
    ! [M2: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na) ) ) ).

% mod_geq
tff(fact_7215_mod__if,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na),M2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na)) ).

% mod_if
tff(fact_7216_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_7217_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_numeral_le_neg_numeral
tff(fact_7218_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_7219_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_7220_neg__numeral__le__numeral,axiom,
    ! [M2: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_le_numeral
tff(fact_7221_neg__numeral__le__numeral,axiom,
    ! [M2: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(Na)) ).

% neg_numeral_le_numeral
tff(fact_7222_neg__numeral__le__numeral,axiom,
    ! [M2: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_le_numeral
tff(fact_7223_neg__numeral__le__numeral,axiom,
    ! [M2: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_le_numeral
tff(fact_7224_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_int != uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)) ).

% zero_neq_neg_numeral
tff(fact_7225_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% zero_neq_neg_numeral
tff(fact_7226_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_neq_neg_numeral
tff(fact_7227_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% zero_neq_neg_numeral
tff(fact_7228_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% zero_neq_neg_numeral
tff(fact_7229_le__mod__geq,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_7230_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_7231_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_7232_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_7233_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_numeral_less_neg_numeral
tff(fact_7234_neg__numeral__less__numeral,axiom,
    ! [M2: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_less_numeral
tff(fact_7235_neg__numeral__less__numeral,axiom,
    ! [M2: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_less_numeral
tff(fact_7236_neg__numeral__less__numeral,axiom,
    ! [M2: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_less_numeral
tff(fact_7237_neg__numeral__less__numeral,axiom,
    ! [M2: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(Na)) ).

% neg_numeral_less_numeral
tff(fact_7238_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xa)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xa)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_7239_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Xa)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_7240_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xa)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_7241_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_7242_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xa)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_7243_one__plus__numeral__commute,axiom,
    ! [Xa: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xa)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xa),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_7244_one__plus__numeral__commute,axiom,
    ! [Xa: num] : plus_p5714425477246183910nteger(one_one_Code_integer,numera6620942414471956472nteger(Xa)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(Xa),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_7245_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,Na) != uminus_uminus_int(one_one_int) ).

% numeral_neq_neg_one
tff(fact_7246_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,Na) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_7247_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,Na) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_7248_numeral__neq__neg__one,axiom,
    ! [Na: num] : numera6620942414471956472nteger(Na) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_7249_numeral__neq__neg__one,axiom,
    ! [Na: num] : numera6690914467698888265omplex(Na) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_7250_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_int != uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)) ).

% one_neq_neg_numeral
tff(fact_7251_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% one_neq_neg_numeral
tff(fact_7252_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% one_neq_neg_numeral
tff(fact_7253_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% one_neq_neg_numeral
tff(fact_7254_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% one_neq_neg_numeral
tff(fact_7255_fact__ge__Suc__0__nat,axiom,
    ! [Na: 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,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_7256_nonzero__norm__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_7257_nonzero__norm__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_7258_power__eq__imp__eq__norm,axiom,
    ! [W2: real,Na: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7259_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,Na: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),Na) = aa_nat_complex(power_power_complex(Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7260_norm__mult__less,axiom,
    ! [Xa: real,R2: real,Ya: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Ya)),S3)
       => 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,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S3)) ) ) ).

% norm_mult_less
tff(fact_7261_norm__mult__less,axiom,
    ! [Xa: complex,R2: real,Ya: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Ya)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S3)) ) ) ).

% norm_mult_less
tff(fact_7262_norm__mult__ineq,axiom,
    ! [Xa: 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,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Ya))) ).

% norm_mult_ineq
tff(fact_7263_norm__mult__ineq,axiom,
    ! [Xa: 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(Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),aa_complex_real(real_V1022390504157884413omplex,Ya))) ).

% norm_mult_ineq
tff(fact_7264_mod__le__divisor,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na)),Na) ) ).

% mod_le_divisor
tff(fact_7265_norm__add__less,axiom,
    ! [Xa: real,R2: real,Ya: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Ya)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_7266_norm__add__less,axiom,
    ! [Xa: complex,R2: real,Ya: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Ya)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_7267_norm__triangle__lt,axiom,
    ! [Xa: real,Ya: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Ya))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya))),E2) ) ).

% norm_triangle_lt
tff(fact_7268_norm__triangle__lt,axiom,
    ! [Xa: complex,Ya: complex,E2: 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,Xa)),aa_complex_real(real_V1022390504157884413omplex,Ya))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xa),Ya))),E2) ) ).

% norm_triangle_lt
tff(fact_7269_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B2: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_7270_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B2: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_7271_norm__triangle__ineq,axiom,
    ! [Xa: 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,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Ya))) ).

% norm_triangle_ineq
tff(fact_7272_norm__triangle__ineq,axiom,
    ! [Xa: 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(Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),aa_complex_real(real_V1022390504157884413omplex,Ya))) ).

% norm_triangle_ineq
tff(fact_7273_norm__triangle__le,axiom,
    ! [Xa: real,Ya: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Ya))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya))),E2) ) ).

% norm_triangle_le
tff(fact_7274_norm__triangle__le,axiom,
    ! [Xa: complex,Ya: complex,E2: 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,Xa)),aa_complex_real(real_V1022390504157884413omplex,Ya))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xa),Ya))),E2) ) ).

% norm_triangle_le
tff(fact_7275_norm__add__leD,axiom,
    ! [A: 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,A),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,A)),C2)) ) ).

% norm_add_leD
tff(fact_7276_norm__add__leD,axiom,
    ! [A: 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(A),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,A)),C2)) ) ).

% norm_add_leD
tff(fact_7277_norm__power__ineq,axiom,
    ! [Xa: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xa)),Na)) ).

% norm_power_ineq
tff(fact_7278_norm__power__ineq,axiom,
    ! [Xa: complex,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xa),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),Na)) ).

% norm_power_ineq
tff(fact_7279_norm__diff__triangle__less,axiom,
    ! [Xa: 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,Xa),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,Xa),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_7280_norm__diff__triangle__less,axiom,
    ! [Xa: 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(Xa),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(Xa),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_7281_norm__triangle__sub,axiom,
    ! [Xa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xa)),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,Xa),Ya)))) ).

% norm_triangle_sub
tff(fact_7282_norm__triangle__sub,axiom,
    ! [Xa: complex,Ya: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),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(Xa),Ya)))) ).

% norm_triangle_sub
tff(fact_7283_norm__triangle__ineq4,axiom,
    ! [A: 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,A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2))) ).

% norm_triangle_ineq4
tff(fact_7284_norm__triangle__ineq4,axiom,
    ! [A: 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(A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2))) ).

% norm_triangle_ineq4
tff(fact_7285_norm__diff__triangle__le,axiom,
    ! [Xa: 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,Xa),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,Xa),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7286_norm__diff__triangle__le,axiom,
    ! [Xa: 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(Xa),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(Xa),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7287_norm__triangle__le__diff,axiom,
    ! [Xa: real,Ya: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,Ya))),E2)
     => 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,Xa),Ya))),E2) ) ).

% norm_triangle_le_diff
tff(fact_7288_norm__triangle__le__diff,axiom,
    ! [Xa: complex,Ya: complex,E2: 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,Xa)),aa_complex_real(real_V1022390504157884413omplex,Ya))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xa),Ya))),E2) ) ).

% norm_triangle_le_diff
tff(fact_7289_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_zero_le_neg_numeral
tff(fact_7290_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_zero_le_neg_numeral
tff(fact_7291_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_zero_le_neg_numeral
tff(fact_7292_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) ).

% not_zero_le_neg_numeral
tff(fact_7293_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_7294_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_7295_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_7296_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_7297_fact__less__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ) ).

% fact_less_mono
tff(fact_7298_fact__less__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ) ).

% fact_less_mono
tff(fact_7299_fact__less__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono
tff(fact_7300_fact__less__mono,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M2)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ) ).

% fact_less_mono
tff(fact_7301_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_7302_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_7303_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_7304_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_7305_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) ).

% not_zero_less_neg_numeral
tff(fact_7306_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_zero_less_neg_numeral
tff(fact_7307_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_zero_less_neg_numeral
tff(fact_7308_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_zero_less_neg_numeral
tff(fact_7309_norm__diff__ineq,axiom,
    ! [A: 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,A)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))) ).

% norm_diff_ineq
tff(fact_7310_norm__diff__ineq,axiom,
    ! [A: 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,A)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))) ).

% norm_diff_ineq
tff(fact_7311_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: rat,C2: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2) = B2,aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_7312_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C2: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = 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,W2)),C2) = B2,aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_7313_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: complex,C2: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2) = B2,numera6690914467698888265omplex(W2) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_7314_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> $ite(C2 != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2),aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_7315_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = aa_num_real(numeral_numeral_real,W2) )
    <=> $ite(C2 != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2),aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_7316_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = numera6690914467698888265omplex(W2) )
    <=> $ite(C2 != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2),numera6690914467698888265omplex(W2) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_7317_div__less__mono,axiom,
    ! [A2: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_7318_norm__triangle__ineq2,axiom,
    ! [A: 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,A)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% norm_triangle_ineq2
tff(fact_7319_norm__triangle__ineq2,axiom,
    ! [A: 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,A)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B2))) ).

% norm_triangle_ineq2
tff(fact_7320_not__one__le__neg__numeral,axiom,
    ! [M2: 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,M2))) ).

% not_one_le_neg_numeral
tff(fact_7321_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).

% not_one_le_neg_numeral
tff(fact_7322_not__one__le__neg__numeral,axiom,
    ! [M2: 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,M2))) ).

% not_one_le_neg_numeral
tff(fact_7323_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))) ).

% not_one_le_neg_numeral
tff(fact_7324_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_7325_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_7326_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_7327_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_7328_neg__numeral__le__neg__one,axiom,
    ! [M2: 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,M2))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_7329_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_7330_neg__numeral__le__neg__one,axiom,
    ! [M2: 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,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_7331_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_7332_neg__one__le__numeral,axiom,
    ! [M2: 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,M2)) ).

% neg_one_le_numeral
tff(fact_7333_neg__one__le__numeral,axiom,
    ! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M2)) ).

% neg_one_le_numeral
tff(fact_7334_neg__one__le__numeral,axiom,
    ! [M2: 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,M2)) ).

% neg_one_le_numeral
tff(fact_7335_neg__one__le__numeral,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(one_one_int)),aa_num_int(numeral_numeral_int,M2)) ).

% neg_one_le_numeral
tff(fact_7336_neg__numeral__le__one,axiom,
    ! [M2: 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,M2))),one_one_real) ).

% neg_numeral_le_one
tff(fact_7337_neg__numeral__le__one,axiom,
    ! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_7338_neg__numeral__le__one,axiom,
    ! [M2: 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,M2))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_7339_neg__numeral__le__one,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),one_one_int) ).

% neg_numeral_le_one
tff(fact_7340_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(one_one_int)),uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_7341_not__neg__one__less__neg__numeral,axiom,
    ! [M2: 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,M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_7342_not__neg__one__less__neg__numeral,axiom,
    ! [M2: 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,M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_7343_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_7344_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))) ).

% not_one_less_neg_numeral
tff(fact_7345_not__one__less__neg__numeral,axiom,
    ! [M2: 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,M2))) ).

% not_one_less_neg_numeral
tff(fact_7346_not__one__less__neg__numeral,axiom,
    ! [M2: 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,M2))) ).

% not_one_less_neg_numeral
tff(fact_7347_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).

% not_one_less_neg_numeral
tff(fact_7348_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_7349_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_7350_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_7351_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_7352_neg__one__less__numeral,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(one_one_int)),aa_num_int(numeral_numeral_int,M2)) ).

% neg_one_less_numeral
tff(fact_7353_neg__one__less__numeral,axiom,
    ! [M2: 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,M2)) ).

% neg_one_less_numeral
tff(fact_7354_neg__one__less__numeral,axiom,
    ! [M2: 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,M2)) ).

% neg_one_less_numeral
tff(fact_7355_neg__one__less__numeral,axiom,
    ! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M2)) ).

% neg_one_less_numeral
tff(fact_7356_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),one_one_int) ).

% neg_numeral_less_one
tff(fact_7357_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real) ).

% neg_numeral_less_one
tff(fact_7358_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_7359_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_7360_mod__eq__nat1E,axiom,
    ! [M2: nat,Q5: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q5) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => ~ ! [S4: nat] : M2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q5),S4)) ) ) ).

% mod_eq_nat1E
tff(fact_7361_mod__eq__nat2E,axiom,
    ! [M2: nat,Q5: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q5) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ~ ! [S4: nat] : Na != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q5),S4)) ) ) ).

% mod_eq_nat2E
tff(fact_7362_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Na: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ya),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
       => ? [Q3: nat] : Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_7363_fact__mod,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,M2)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7364_fact__mod,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,M2)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7365_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7366_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7367_fact__le__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7368_fact__le__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7369_fact__le__power,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7370_fact__le__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7371_norm__exp,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Xa))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,Xa))) ).

% norm_exp
tff(fact_7372_norm__exp,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Xa))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,Xa))) ).

% norm_exp
tff(fact_7373_powr__neg__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Na))) ) ) ).

% powr_neg_numeral
tff(fact_7374_fact__diff__Suc,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M2))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na))) ) ) ).

% fact_diff_Suc
tff(fact_7375_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_7376_power__eq__1__iff,axiom,
    ! [W2: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Na) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7377_power__eq__1__iff,axiom,
    ! [W2: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),Na) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7378_norm__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C2: real,D: 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,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7379_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B2: complex,C2: complex,D: 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(A),B2)),aa_complex_complex(plus_plus_complex(C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B2),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7380_norm__sgn,axiom,
    ! [Xa: real] :
      aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,Xa)) = $ite(Xa = zero_zero_real,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_7381_norm__sgn,axiom,
    ! [Xa: complex] :
      aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,Xa)) = $ite(Xa = zero_zero_complex,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_7382_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_7383_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W2: 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,W2))
    <=> $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,W2)),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,W2)),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_7384_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_7385_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),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,W2)),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,W2)),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_7386_split__mod,axiom,
    ! [P: fun_nat_o,M2: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,M2) )
        & ( ( Na != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I4)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_7387_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> $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,W2))),C2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7388_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(C2 != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7389_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> $ite(C2 != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7390_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = 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,W2))),C2) = B2,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7391_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: rat,C2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2) = B2,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7392_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: complex,C2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2) = B2,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7393_norm__triangle__ineq3,axiom,
    ! [A: 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,A)),aa_real_real(real_V7735802525324610683m_real,B2)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% norm_triangle_ineq3
tff(fact_7394_norm__triangle__ineq3,axiom,
    ! [A: 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,A)),aa_complex_real(real_V1022390504157884413omplex,B2)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B2))) ).

% norm_triangle_ineq3
tff(fact_7395_nat__mod__distrib,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Ya)) ) ) ) ).

% nat_mod_distrib
tff(fact_7396_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),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,W2)),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,W2)),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_7397_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_7398_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W2: 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,W2))
    <=> $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,W2)),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,W2)),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_7399_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_7400_Suc__times__mod__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2)
     => ( 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,M2),Na))),M2) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_7401_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),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,W2))),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,W2))),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_7402_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_7403_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W2: 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,W2)))
    <=> $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,W2))),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,W2))),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,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_7404_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B2),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_7405_norm__inverse__le__norm,axiom,
    ! [R2: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,Xa))),aa_real_real(inverse_inverse_real,R2)) ) ) ).

% norm_inverse_le_norm
tff(fact_7406_norm__inverse__le__norm,axiom,
    ! [R2: real,Xa: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,Xa))),aa_real_real(inverse_inverse_real,R2)) ) ) ).

% norm_inverse_le_norm
tff(fact_7407_CauchyD,axiom,
    ! [X3: fun_nat_complex,E2: real] :
      ( topolo6517432010174082258omplex(X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ? [M9: nat] :
          ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M3)
           => ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N3)
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X3,M3)),aa_nat_complex(X3,N3)))),E2) ) ) ) ) ).

% CauchyD
tff(fact_7408_CauchyD,axiom,
    ! [X3: fun_nat_real,E2: real] :
      ( topolo4055970368930404560y_real(X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ? [M9: nat] :
          ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M3)
           => ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N3)
               => 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,aa_nat_real(X3,M3)),aa_nat_real(X3,N3)))),E2) ) ) ) ) ).

% CauchyD
tff(fact_7409_CauchyI,axiom,
    ! [X3: fun_nat_complex] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => ? [M10: nat] :
            ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M10),M4)
             => ! [N2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M10),N2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X3,M4)),aa_nat_complex(X3,N2)))),E) ) ) )
     => topolo6517432010174082258omplex(X3) ) ).

% CauchyI
tff(fact_7410_CauchyI,axiom,
    ! [X3: fun_nat_real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => ? [M10: nat] :
            ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M10),M4)
             => ! [N2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M10),N2)
                 => 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,aa_nat_real(X3,M4)),aa_nat_real(X3,N2)))),E) ) ) )
     => topolo4055970368930404560y_real(X3) ) ).

% CauchyI
tff(fact_7411_Cauchy__iff,axiom,
    ! [X3: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X3)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [M8: nat] :
            ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X3,M)),aa_nat_complex(X3,N)))),E3) ) ) ) ) ).

% Cauchy_iff
tff(fact_7412_Cauchy__iff,axiom,
    ! [X3: fun_nat_real] :
      ( topolo4055970368930404560y_real(X3)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [M8: nat] :
            ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N)
                 => 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,aa_nat_real(X3,M)),aa_nat_real(X3,N)))),E3) ) ) ) ) ).

% Cauchy_iff
tff(fact_7413_assms_I2_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,x),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),n)) ).

% assms(2)
tff(fact_7414_enat__ord__number_I1_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(1)
tff(fact_7415_enat__ord__number_I2_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(2)
tff(fact_7416_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K4: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K4)
       => ( 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))),K4)
         => 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)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(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,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7417_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K4: real,Na: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K4)
       => ( 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))),K4)
         => 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)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(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,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7418_bounded__linear__axioms__def,axiom,
    ! [F: fun_complex_complex] :
      ( real_V7139242839884736329omplex(F)
    <=> ? [K5: real] :
        ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5)) ) ).

% bounded_linear_axioms_def
tff(fact_7419_bounded__linear__axioms_Ointro,axiom,
    ! [F: fun_complex_complex] :
      ( ? [K6: real] :
        ! [X2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X2)),K6))
     => real_V7139242839884736329omplex(F) ) ).

% bounded_linear_axioms.intro
tff(fact_7420_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_7421_pow__sum,axiom,
    ! [A: 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,A),B2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ).

% pow_sum
tff(fact_7422_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Na: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xa)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% member_bound
tff(fact_7423_valid__pres__insert,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_insert(Ta,Xa)),Na) ) ) ).

% valid_pres_insert
tff(fact_7424_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat,Ya: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Ya)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_7425_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_7426_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat,Ya: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xa)),Ya)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Ya)
              | ( Xa = Ya ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_7427_semiring__norm_I78_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M2)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% semiring_norm(78)
tff(fact_7428_semiring__norm_I71_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M2)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% semiring_norm(71)
tff(fact_7429_semiring__norm_I75_J,axiom,
    ! [M2: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M2),one) ).

% semiring_norm(75)
tff(fact_7430_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Na) ).

% semiring_norm(68)
tff(fact_7431_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_7432_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera6690914467698888265omplex(Na) = one_one_complex )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7433_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Na) = one_one_rat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7434_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_real(numeral_numeral_real,Na) = one_one_real )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7435_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Na) = one_one_nat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7436_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_int(numeral_numeral_int,Na) = one_one_int )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7437_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera1916890842035813515d_enat(Na) = one_on7984719198319812577d_enat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7438_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera6620942414471956472nteger(Na) = one_one_Code_integer )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_7439_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7440_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7441_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7442_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7443_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7444_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_on7984719198319812577d_enat = numera1916890842035813515d_enat(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7445_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_Code_integer = numera6620942414471956472nteger(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_7446_num__double,axiom,
    ! [Na: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),Na) = aa_num_num(bit0,Na) ).

% num_double
tff(fact_7447_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Na)) ).

% semiring_norm(76)
tff(fact_7448_semiring__norm_I69_J,axiom,
    ! [M2: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M2)),one) ).

% semiring_norm(69)
tff(fact_7449_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( uminus_uminus_int(one_one_int) = uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7450_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7451_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7452_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7453_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7454_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)) = uminus_uminus_int(one_one_int) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7455_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7456_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7457_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7458_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7459_Suc__numeral,axiom,
    ! [Na: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Na),one)) ).

% Suc_numeral
tff(fact_7460_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: 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,M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7461_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7462_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: 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,M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7463_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(one_one_int)),uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7464_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(one_one_int))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7465_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7466_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7467_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7468_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_7469_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_7470_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_7471_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_7472_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_7473_one__add__one,axiom,
    plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_7474_one__add__one,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_7475_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_7476_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_7477_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_7478_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_7479_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_7480_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc'
tff(fact_7481_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc
tff(fact_7482_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_7483_numeral__plus__one,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7484_numeral__plus__one,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7485_numeral__plus__one,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7486_numeral__plus__one,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7487_numeral__plus__one,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7488_numeral__plus__one,axiom,
    ! [Na: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7489_numeral__plus__one,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(Na),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(Na),one)) ).

% numeral_plus_one
tff(fact_7490_one__plus__numeral,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7491_one__plus__numeral,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7492_one__plus__numeral,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7493_one__plus__numeral,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7494_one__plus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7495_one__plus__numeral,axiom,
    ! [Na: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7496_one__plus__numeral,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(one_one_Code_integer,numera6620942414471956472nteger(Na)) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Na)) ).

% one_plus_numeral
tff(fact_7497_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_7498_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_7499_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Na)),one_one_Code_integer)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_7500_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_7501_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_7502_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_7503_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_7504_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_7505_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_7506_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_7507_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_7508_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),numera6620942414471956472nteger(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_7509_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_7510_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_7511_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_7512_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_7513_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_7514_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_7515_power2__eq__iff__nonneg,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_7516_power2__eq__iff__nonneg,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => ( ( aa_nat_rat(power_power_rat(Xa),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))) )
        <=> ( Xa = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_7517_power2__eq__iff__nonneg,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xa = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_7518_power2__eq__iff__nonneg,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( ( aa_nat_int(power_power_int(Xa),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))) )
        <=> ( Xa = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_7519_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_7520_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_7521_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_7522_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(one_one_int)),uminus_uminus_int(one_one_int)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7523_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_7524_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7525_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7526_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_7527_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_7528_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_7529_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_7530_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(one_one_int)),one_one_int) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7531_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_7532_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7533_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7534_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_7535_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),uminus_uminus_int(one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7536_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_7537_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7538_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7539_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_7540_sum__power2__eq__zero__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_7541_sum__power2__eq__zero__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_7542_sum__power2__eq__zero__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xa = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_7543_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_7544_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_7545_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_7546_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_7547_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_7548_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_7549_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_7550_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_7551_diff__numeral__special_I3_J,axiom,
    ! [Na: 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,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_7552_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_7553_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_7554_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Na)) ).

% diff_numeral_special(3)
tff(fact_7555_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),one_one_int) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M2),one))) ).

% diff_numeral_special(4)
tff(fact_7556_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M2),one))) ).

% diff_numeral_special(4)
tff(fact_7557_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M2),one))) ).

% diff_numeral_special(4)
tff(fact_7558_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(M2),one))) ).

% diff_numeral_special(4)
tff(fact_7559_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M2),one))) ).

% diff_numeral_special(4)
tff(fact_7560_add__self__mod__2,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),M2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_7561_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_7562_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_7563_one__less__floor,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa) ) ).

% one_less_floor
tff(fact_7564_one__less__floor,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,Xa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa) ) ).

% one_less_floor
tff(fact_7565_floor__le__one,axiom,
    ! [Xa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_7566_floor__le__one,axiom,
    ! [Xa: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_7567_mod2__gr__0,axiom,
    ! [M2: 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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_7568_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Ya: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Z),Ya)
     => ( plus_p3455044024723400733d_enat(Xa,minus_3235023915231533773d_enat(Ya,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xa,Ya),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_7569_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Na),zero_z5237406670263579293d_enat)
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_7570_i0__lb,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),Na) ).

% i0_lb
tff(fact_7571_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),one)
    <=> ( Xa = one ) ) ).

% le_num_One_iff
tff(fact_7572_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,M3),N2)
             => aa_Extended_enat_o(P,M3) )
         => aa_Extended_enat_o(P,N2) )
     => aa_Extended_enat_o(P,Na) ) ).

% enat_less_induct
tff(fact_7573_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_7574_add__One__commute,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(one),Na) = aa_num_num(plus_plus_num(Na),one) ).

% add_One_commute
tff(fact_7575_enat__0__less__mult__iff,axiom,
    ! [M2: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M2,Na))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),M2)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_7576_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_7577_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_7578_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_7579_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_7580_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_7581_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_7582_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_7583_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_7584_less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% less_exp
tff(fact_7585_self__le__ge2__pow,axiom,
    ! [K: nat,M2: 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,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M2)) ) ).

% self_le_ge2_pow
tff(fact_7586_power2__nat__le__eq__le,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% power2_nat_le_eq_le
tff(fact_7587_power2__nat__le__imp__le,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% power2_nat_le_imp_le
tff(fact_7588_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_7589_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_7590_power2__le__imp__le,axiom,
    ! [Xa: 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,Xa),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,Xa),Ya) ) ) ).

% power2_le_imp_le
tff(fact_7591_power2__le__imp__le,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(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,Xa),Ya) ) ) ).

% power2_le_imp_le
tff(fact_7592_power2__le__imp__le,axiom,
    ! [Xa: 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,Xa),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,Xa),Ya) ) ) ).

% power2_le_imp_le
tff(fact_7593_power2__le__imp__le,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(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,Xa),Ya) ) ) ).

% power2_le_imp_le
tff(fact_7594_power2__eq__imp__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => ( Xa = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_7595_power2__eq__imp__eq,axiom,
    ! [Xa: rat,Ya: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(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),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
         => ( Xa = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_7596_power2__eq__imp__eq,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
         => ( Xa = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_7597_power2__eq__imp__eq,axiom,
    ! [Xa: int,Ya: int] :
      ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(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),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
         => ( Xa = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_7598_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_7599_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_7600_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_7601_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_7602_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_7603_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_7604_left__add__twice,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B2) ).

% left_add_twice
tff(fact_7605_left__add__twice,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),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))),A)),B2) ).

% left_add_twice
tff(fact_7606_left__add__twice,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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))),A)),B2) ).

% left_add_twice
tff(fact_7607_left__add__twice,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B2) ).

% left_add_twice
tff(fact_7608_left__add__twice,axiom,
    ! [A: extended_enat,B2: extended_enat] : plus_p3455044024723400733d_enat(A,plus_p3455044024723400733d_enat(A,B2)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),B2) ).

% left_add_twice
tff(fact_7609_left__add__twice,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A),B2) ).

% left_add_twice
tff(fact_7610_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_7611_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_7612_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_7613_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_7614_mult__2__right,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(Z,numera1916890842035813515d_enat(aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2_right
tff(fact_7615_mult__2__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2_right
tff(fact_7616_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_7617_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_7618_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_7619_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_7620_mult__2,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Z) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2
tff(fact_7621_mult__2,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Z) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2
tff(fact_7622_abs__le__square__iff,axiom,
    ! [Xa: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),abs_abs_Code_integer(Ya))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xa,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_7623_abs__le__square__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7624_abs__le__square__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),aa_rat_rat(abs_abs_rat,Ya))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7625_abs__le__square__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),aa_int_int(abs_abs_int,Ya))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7626_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_7627_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_7628_card__2__iff,axiom,
    ! [S: set_complex] :
      ( ( finite_card_complex(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: complex,Y4: complex] :
          ( ( S = insert_complex(X,insert_complex(Y4,bot_bot_set_complex)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7629_card__2__iff,axiom,
    ! [S: set_Product_unit] :
      ( ( finite410649719033368117t_unit(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: product_unit,Y4: product_unit] :
          ( ( S = insert_Product_unit(X,insert_Product_unit(Y4,bot_bo3957492148770167129t_unit)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7630_card__2__iff,axiom,
    ! [S: set_list_nat] :
      ( ( finite_card_list_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: list_nat,Y4: list_nat] :
          ( ( S = insert_list_nat(X,insert_list_nat(Y4,bot_bot_set_list_nat)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7631_card__2__iff,axiom,
    ! [S: set_nat] :
      ( ( finite_card_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: nat,Y4: nat] :
          ( ( S = insert_nat(X,insert_nat(Y4,bot_bot_set_nat)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7632_card__2__iff,axiom,
    ! [S: set_int] :
      ( ( finite_card_int(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: int,Y4: int] :
          ( ( S = insert_int(X,insert_int(Y4,bot_bot_set_int)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7633_card__2__iff,axiom,
    ! [S: set_o] :
      ( ( finite_card_o(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: $o,Y4: $o] :
          ( ( S = insert_o((X),insert_o((Y4),bot_bot_set_o)) )
          & ~ ( (X)
            <=> (Y4) ) ) ) ).

% card_2_iff
tff(fact_7634_card__2__iff,axiom,
    ! [S: set_set_nat] :
      ( ( finite_card_set_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: set_nat,Y4: set_nat] :
          ( ( S = insert_set_nat(X,insert_set_nat(Y4,bot_bot_set_set_nat)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7635_card__2__iff,axiom,
    ! [S: set_real] :
      ( ( finite_card_real(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: real,Y4: real] :
          ( ( S = insert_real(X,insert_real(Y4,bot_bot_set_real)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7636_card__2__iff,axiom,
    ! [S: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X: extended_enat,Y4: extended_enat] :
          ( ( S = insert_Extended_enat(X,insert_Extended_enat(Y4,bot_bo7653980558646680370d_enat)) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_7637_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_7638_nat__induct2,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(P,N2)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct2
tff(fact_7639_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% two_realpow_ge_one
tff(fact_7640_square__fact__le__2__fact,axiom,
    ! [Na: 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,Na)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% square_fact_le_2_fact
tff(fact_7641_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_7642_diff__le__diff__pow,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_7643_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_7644_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_7645_power2__less__imp__less,axiom,
    ! [Xa: 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,Xa),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,Xa),Ya) ) ) ).

% power2_less_imp_less
tff(fact_7646_power2__less__imp__less,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(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,Xa),Ya) ) ) ).

% power2_less_imp_less
tff(fact_7647_power2__less__imp__less,axiom,
    ! [Xa: 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,Xa),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,Xa),Ya) ) ) ).

% power2_less_imp_less
tff(fact_7648_power2__less__imp__less,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(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,Xa),Ya) ) ) ).

% power2_less_imp_less
tff(fact_7649_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_7650_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_7651_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_7652_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_7653_sum__power2__le__zero__iff,axiom,
    ! [Xa: 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,Xa),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)
    <=> ( ( Xa = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_7654_sum__power2__le__zero__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_7655_sum__power2__le__zero__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_7656_sum__power2__ge__zero,axiom,
    ! [Xa: 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,Xa),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_7657_sum__power2__ge__zero,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_7658_sum__power2__ge__zero,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_7659_not__sum__power2__lt__zero,axiom,
    ! [Xa: 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,Xa),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_7660_not__sum__power2__lt__zero,axiom,
    ! [Xa: rat,Ya: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_7661_not__sum__power2__lt__zero,axiom,
    ! [Xa: int,Ya: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_7662_sum__power2__gt__zero__iff,axiom,
    ! [Xa: 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,Xa),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)))))
    <=> ( ( Xa != zero_zero_real )
        | ( Ya != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_7663_sum__power2__gt__zero__iff,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_rat )
        | ( Ya != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_7664_sum__power2__gt__zero__iff,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xa != zero_zero_int )
        | ( Ya != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_7665_field__less__half__sum,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_7666_field__less__half__sum,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_7667_square__le__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_7668_square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xa,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_7669_square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_7670_square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(one_one_int)),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_7671_power2__le__iff__abs__le,axiom,
    ! [Ya: code_integer,Xa: 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(Xa,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(Xa)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_7672_power2__le__iff__abs__le,axiom,
    ! [Ya: real,Xa: 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,Xa),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,Xa)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_7673_power2__le__iff__abs__le,axiom,
    ! [Ya: rat,Xa: 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(Xa),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,Xa)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_7674_power2__le__iff__abs__le,axiom,
    ! [Ya: int,Xa: 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(Xa),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,Xa)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_7675_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7676_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,semiri4939895301339042750nteger(Na)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) ).

% of_nat_less_two_power
tff(fact_7677_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7678_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7679_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_7680_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_7681_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,Na: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_7682_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_7683_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_7684_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,Na: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_7685_zero__le__even__power_H,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_7686_zero__le__even__power_H,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_7687_zero__le__even__power_H,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_7688_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_7689_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M2: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_7690_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M2: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na) != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_7691_abs__square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xa,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_7692_abs__square__le__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7693_abs__square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_7694_abs__square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7695_abs__square__less__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(Xa,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xa)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_7696_abs__square__less__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7697_abs__square__less__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xa)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_7698_abs__square__less__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xa)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7699_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => 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))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa_nat_o(P,N2)
             => 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))),N2))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_bit_induct
tff(fact_7700_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_7701_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_7702_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C2: real,B2: real,D: 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,A),C2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D))),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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(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_7703_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_Bit0
tff(fact_7704_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na)) ).

% numeral_Bit0
tff(fact_7705_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_Bit0
tff(fact_7706_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na)) ).

% numeral_Bit0
tff(fact_7707_numeral__Bit0,axiom,
    ! [Na: num] : numera1916890842035813515d_enat(aa_num_num(bit0,Na)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),numera1916890842035813515d_enat(Na)) ).

% numeral_Bit0
tff(fact_7708_numeral__Bit0,axiom,
    ! [Na: num] : numera6620942414471956472nteger(aa_num_num(bit0,Na)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Na)) ).

% numeral_Bit0
tff(fact_7709_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_7710_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_7711_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_7712_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_7713_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_7714_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_7715_mult__numeral__1,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(one),A) = A ).

% mult_numeral_1
tff(fact_7716_mult__numeral__1,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(one),A) = A ).

% mult_numeral_1
tff(fact_7717_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_7718_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_7719_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_7720_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_7721_mult__numeral__1__right,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(one)) = A ).

% mult_numeral_1_right
tff(fact_7722_mult__numeral__1__right,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(A,numera6620942414471956472nteger(one)) = A ).

% mult_numeral_1_right
tff(fact_7723_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_7724_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_7725_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_7726_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_7727_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_7728_numeral__One,axiom,
    numera1916890842035813515d_enat(one) = one_on7984719198319812577d_enat ).

% numeral_One
tff(fact_7729_numeral__One,axiom,
    numera6620942414471956472nteger(one) = one_one_Code_integer ).

% numeral_One
tff(fact_7730_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_7731_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_7732_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_7733_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),Na) ).

% Suc_nat_number_of_add
tff(fact_7734_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).

% inverse_numeral_1
tff(fact_7735_sum__squares__bound,axiom,
    ! [Xa: 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))),Xa)),Ya)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_7736_sum__squares__bound,axiom,
    ! [Xa: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_7737_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),B2)
       => ( modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)) = modulo364778990260209775nteger(A,B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_7738_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A: 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(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_7739_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A: 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,A),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,A),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,A),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_7740_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_7741_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_7742_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_7743_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7744_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7745_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_7746_odd__power__less__zero,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_7747_odd__power__less__zero,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_7748_odd__power__less__zero,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_7749_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)
       => ? [N2: 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),N2)),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,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_7750_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)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N2)),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,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_7751_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))) ) ).

% plus_inverse_ge_2
tff(fact_7752_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),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(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7753_exp__bound__half,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(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_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7754_less__log2__of__power,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))) ) ).

% less_log2_of_power
tff(fact_7755_le__log2__of__power,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))) ) ).

% le_log2_of_power
tff(fact_7756_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),B2)
       => ( times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))) = divide6298287555418463151nteger(A,B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_7757_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A: 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(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_7758_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A: 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,A),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,A),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,A),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_7759_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_7760_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_7761_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_7762_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q5: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q5))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q5)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_7763_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q5: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q5))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q5)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_7764_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q5: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q5))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q5)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_7765_num_Osize_I5_J,axiom,
    ! [X23: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_7766_log2__of__power__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_7767_exp__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_7768_pos__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_7769_neg__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_7770_pos__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_7771_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_7772_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,one))),B2) = uminus_uminus_int(B2) ).

% mult_1s_ring_1(1)
tff(fact_7773_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_7774_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B2) = aa_rat_rat(uminus_uminus_rat,B2) ).

% mult_1s_ring_1(1)
tff(fact_7775_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)),B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ).

% mult_1s_ring_1(1)
tff(fact_7776_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_7777_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),uminus_uminus_int(aa_num_int(numeral_numeral_int,one))) = uminus_uminus_int(B2) ).

% mult_1s_ring_1(2)
tff(fact_7778_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_7779_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] : aa_rat_rat(aa_rat_fun_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_7780_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] : times_3573771949741848930nteger(B2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ).

% mult_1s_ring_1(2)
tff(fact_7781_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_7782_uminus__numeral__One,axiom,
    uminus_uminus_int(aa_num_int(numeral_numeral_int,one)) = uminus_uminus_int(one_one_int) ).

% uminus_numeral_One
tff(fact_7783_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_7784_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_7785_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_7786_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_7787_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_7788_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_7789_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_7790_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_7791_arith__geo__mean,axiom,
    ! [U: real,Xa: 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,Xa),Ya) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa),Ya)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_7792_arith__geo__mean,axiom,
    ! [U: rat,Xa: rat,Ya: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Ya) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_7793_mod__double__modulus,axiom,
    ! [M2: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
       => ( ( modulo364778990260209775nteger(Xa,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2)) = modulo364778990260209775nteger(Xa,M2) )
          | ( modulo364778990260209775nteger(Xa,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xa,M2),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_7794_mod__double__modulus,axiom,
    ! [M2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M2) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_7795_mod__double__modulus,axiom,
    ! [M2: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),M2) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_7796_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)),B2) = modulo364778990260209775nteger(A,B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_7797_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),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,A),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,A),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,A),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_7798_divmod__digit__1_I2_J,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_7799_log2__of__power__le,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
       => 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,M2))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_7800_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),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(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_7801_exp__bound__lemma,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(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_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_7802_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(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,Xa)),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))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_7803_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7804_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_7805_neg__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_7806_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_7807_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_7808_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7809_arctan__double,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xa)) = aa_real_real(arctan,aa_real_real(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))),Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7810_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)))
         => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),one_one_Code_integer) = divide6298287555418463151nteger(A,B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_7811_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),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,A),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,A),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,A),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_7812_divmod__digit__1_I1_J,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_7813_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(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,Xa)),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,Xa),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),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_7814_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(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),Xa))),Xa))),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_7815_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_7816_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_7817_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(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,Na),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_7818_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_real_real(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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_7819_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_7820_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_7821_inrange,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(Ta)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ) ).

% inrange
tff(fact_7822_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xa: code_integer] :
      ( ! [X2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X2),power_8256067586552552935nteger(X2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xa)),power_8256067586552552935nteger(Xa,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7823_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xa: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
         => aa_real_o(aa_real_fun_real_o(P,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7824_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xa: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X2)
         => aa_rat_o(aa_rat_fun_rat_o(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xa)),aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7825_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xa: int] :
      ( ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X2)
         => aa_int_o(aa_int_fun_int_o(P,X2),aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xa)),aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7826_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_7827_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_7828_set__bit__0,axiom,
    ! [A: int] : bit_se7879613467334960850it_int(zero_zero_nat,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_7829_low__inv,axiom,
    ! [Xa: nat,Na: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Xa)),Na) = Xa ) ) ).

% low_inv
tff(fact_7830_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,M2: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na) ) ) ).

% set_n_deg_not_0
tff(fact_7831_high__inv,axiom,
    ! [Xa: nat,Na: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Xa)),Na) = Ya ) ) ).

% high_inv
tff(fact_7832_bit__split__inv,axiom,
    ! [Xa: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),D)),D) = Xa ).

% bit_split_inv
tff(fact_7833_high__def,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% high_def
tff(fact_7834_low__def,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% low_def
tff(fact_7835_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ) ).

% high_bound_aux
tff(fact_7836_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
          & ~ aa_set_int_o(ord_less_eq_set_int(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_7837_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_7838_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_7839_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_7840_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_7841_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_7842_atLeastAtMost__iff,axiom,
    ! [I: extended_enat,L: extended_enat,U: extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(I),set_or5403411693681687835d_enat(L,U))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,L),I)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7843_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7844_atLeastAtMost__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( aa_set_o_o(member_o((I)),set_or8904488021354931149Most_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(I))
        & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(I)),(U)) ) ) ).

% atLeastAtMost_iff
tff(fact_7845_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(ord_less_eq_set_int(L),I)
        & aa_set_int_o(ord_less_eq_set_int(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7846_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_7847_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_7848_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7849_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_7850_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_7851_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_7852_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_7853_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_7854_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_7855_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,set_Extended_enat2(Xs)) ).

% List.finite_set
tff(fact_7856_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se7879613467334960850it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_7857_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se7879613467334960850it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_7858_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or1269000886237332187st_nat(L,U)) ).

% finite_atLeastAtMost
tff(fact_7859_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B2: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B2)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_7860_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B2) )
    <=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7861_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = set_or5403411693681687835d_enat(A,B2) )
    <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7862_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B2) )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7863_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B2) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7864_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B2: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B2) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7865_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A,B2) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7866_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B2: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B2) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7867_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B2: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B2) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_7868_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B2: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B2)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_7869_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B2) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7870_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B2) = bot_bo7653980558646680370d_enat )
    <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7871_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( set_or370866239135849197et_int(A,B2) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7872_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( set_or633870826150836451st_rat(A,B2) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7873_atLeastatMost__empty__iff,axiom,
    ! [A: num,B2: num] :
      ( ( set_or7049704709247886629st_num(A,B2) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7874_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( set_or1269000886237332187st_nat(A,B2) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7875_atLeastatMost__empty__iff,axiom,
    ! [A: int,B2: int] :
      ( ( set_or1266510415728281911st_int(A,B2) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7876_atLeastatMost__empty__iff,axiom,
    ! [A: real,B2: real] :
      ( ( set_or1222579329274155063t_real(A,B2) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_7877_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B2: set_int,C2: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B2),set_or370866239135849197et_int(C2,D))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B2)
        | ( aa_set_int_o(ord_less_eq_set_int(C2),A)
          & aa_set_int_o(ord_less_eq_set_int(B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7878_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B2)),set_or633870826150836451st_rat(C2,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7879_atLeastatMost__subset__iff,axiom,
    ! [A: num,B2: num,C2: num,D: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B2)),set_or7049704709247886629st_num(C2,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7880_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B2)),set_or1269000886237332187st_nat(C2,D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7881_atLeastatMost__subset__iff,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B2)),set_or1266510415728281911st_int(C2,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7882_atLeastatMost__subset__iff,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B2)),set_or1222579329274155063t_real(C2,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7883_atLeastatMost__empty,axiom,
    ! [B2: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_o,(B2)),(A))
     => ( set_or8904488021354931149Most_o((A),(B2)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_7884_atLeastatMost__empty,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B2),A)
     => ( set_or4548717258645045905et_nat(A,B2) = bot_bot_set_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_7885_atLeastatMost__empty,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B2),A)
     => ( set_or5403411693681687835d_enat(A,B2) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_7886_atLeastatMost__empty,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( set_or633870826150836451st_rat(A,B2) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_7887_atLeastatMost__empty,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( set_or7049704709247886629st_num(A,B2) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_7888_atLeastatMost__empty,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( set_or1269000886237332187st_nat(A,B2) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_7889_atLeastatMost__empty,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( set_or1266510415728281911st_int(A,B2) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_7890_atLeastatMost__empty,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( set_or1222579329274155063t_real(A,B2) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_7891_infinite__Icc__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% infinite_Icc_iff
tff(fact_7892_infinite__Icc__iff,axiom,
    ! [A: real,B2: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% infinite_Icc_iff
tff(fact_7893_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B2: $o,C2: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B2)) = insert_o((C2),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B2) )
        & ( (B2)
        <=> (C2) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7894_atLeastAtMost__singleton__iff,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B2) = insert_set_nat(C2,bot_bot_set_set_nat) )
    <=> ( ( A = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7895_atLeastAtMost__singleton__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C2: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B2) = insert_Extended_enat(C2,bot_bo7653980558646680370d_enat) )
    <=> ( ( A = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7896_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( set_or1269000886237332187st_nat(A,B2) = insert_nat(C2,bot_bot_set_nat) )
    <=> ( ( A = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7897_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( set_or1266510415728281911st_int(A,B2) = insert_int(C2,bot_bot_set_int) )
    <=> ( ( A = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7898_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B2: real,C2: real] :
      ( ( set_or1222579329274155063t_real(A,B2) = insert_real(C2,bot_bot_set_real) )
    <=> ( ( A = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_7899_atLeastAtMost__singleton,axiom,
    ! [A: $o] : set_or8904488021354931149Most_o((A),(A)) = insert_o((A),bot_bot_set_o) ).

% atLeastAtMost_singleton
tff(fact_7900_atLeastAtMost__singleton,axiom,
    ! [A: set_nat] : set_or4548717258645045905et_nat(A,A) = insert_set_nat(A,bot_bot_set_set_nat) ).

% atLeastAtMost_singleton
tff(fact_7901_atLeastAtMost__singleton,axiom,
    ! [A: extended_enat] : set_or5403411693681687835d_enat(A,A) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ).

% atLeastAtMost_singleton
tff(fact_7902_atLeastAtMost__singleton,axiom,
    ! [A: nat] : set_or1269000886237332187st_nat(A,A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_7903_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_7904_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_7905_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_7906_finite__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_7907_finite__list,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_7908_finite__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_7909_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ? [Xs2: list_Extended_enat] : set_Extended_enat2(Xs2) = A2 ) ).

% finite_list
tff(fact_7910_subset__code_I1_J,axiom,
    ! [Xs: list_Extended_enat,B: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),B)
    <=> ! [X: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X),set_Extended_enat2(Xs))
         => aa_set3337841544853333262enat_o(member_Extended_enat(X),B) ) ) ).

% subset_code(1)
tff(fact_7911_subset__code_I1_J,axiom,
    ! [Xs: list_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_set_real_o(member_real(X),B) ) ) ).

% subset_code(1)
tff(fact_7912_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B)
    <=> ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
         => aa_set_set_nat_o(member_set_nat(X),B) ) ) ).

% subset_code(1)
tff(fact_7913_subset__code_I1_J,axiom,
    ! [Xs: list_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(set_o2(Xs)),B)
    <=> ! [X: $o] :
          ( aa_set_o_o(member_o((X)),set_o2(Xs))
         => aa_set_o_o(member_o((X)),B) ) ) ).

% subset_code(1)
tff(fact_7914_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B)
    <=> ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X,B) ) ) ).

% subset_code(1)
tff(fact_7915_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),B)
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X),B) ) ) ).

% subset_code(1)
tff(fact_7916_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B)
    <=> ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_set_int_o(member_int(X),B) ) ) ).

% subset_code(1)
tff(fact_7917_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_se7879613467334960850it_int(Na,K)) ).

% set_bit_greater_eq
tff(fact_7918_infinite__Icc,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B2)) ) ).

% infinite_Icc
tff(fact_7919_infinite__Icc,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B2)) ) ).

% infinite_Icc
tff(fact_7920_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: $o,M2: $o,U: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(M2))
     => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M2)),(U))
       => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_or8904488021354931149Most_o((L),(M2))),set_or8904488021354931149Most_o((M2),(U))) = set_or8904488021354931149Most_o((L),(U)) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7921_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M2),U)
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,M2),set_or633870826150836451st_rat(M2,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7922_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M2),set_or7049704709247886629st_num(M2,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7923_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or1269000886237332187st_nat(L,M2)),set_or1269000886237332187st_nat(M2,U)) = set_or1269000886237332187st_nat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7924_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or1266510415728281911st_int(L,M2)),set_or1266510415728281911st_int(M2,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7925_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,M2: real,U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),M2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M2),U)
       => ( sup_sup_set_real(set_or1222579329274155063t_real(L,M2),set_or1222579329274155063t_real(M2,U)) = set_or1222579329274155063t_real(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7926_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B2: $o] :
      ( ( (A)
      <=> (B2) )
     => ( set_or8904488021354931149Most_o((A),(B2)) = insert_o((A),bot_bot_set_o) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7927_atLeastAtMost__singleton_H,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ( set_or4548717258645045905et_nat(A,B2) = insert_set_nat(A,bot_bot_set_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7928_atLeastAtMost__singleton_H,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A = B2 )
     => ( set_or5403411693681687835d_enat(A,B2) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7929_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
     => ( set_or1269000886237332187st_nat(A,B2) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7930_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
     => ( set_or1266510415728281911st_int(A,B2) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7931_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
     => ( set_or1222579329274155063t_real(A,B2) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_7932_all__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
         => aa_nat_o(P,M) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or1269000886237332187st_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less
tff(fact_7933_ex__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(P,M) )
    <=> ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or1269000886237332187st_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less
tff(fact_7934_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B2: set_int,C2: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B2),set_or370866239135849197et_int(C2,D))
    <=> ( ( ~ aa_set_int_o(ord_less_eq_set_int(A),B2)
          | ( aa_set_int_o(ord_less_eq_set_int(C2),A)
            & aa_set_int_o(ord_less_eq_set_int(B2),D)
            & ( aa_set_int_o(ord_less_set_int(C2),A)
              | aa_set_int_o(ord_less_set_int(B2),D) ) ) )
        & aa_set_int_o(ord_less_eq_set_int(C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7935_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B2: rat,C2: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B2)),set_or633870826150836451st_rat(C2,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7936_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B2: num,C2: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B2)),set_or7049704709247886629st_num(C2,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B2),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7937_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B2: nat,C2: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,set_or1269000886237332187st_nat(A,B2)),set_or1269000886237332187st_nat(C2,D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7938_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B2: int,C2: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B2)),set_or1266510415728281911st_int(C2,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B2),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7939_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B2: real,C2: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B2)),set_or1222579329274155063t_real(C2,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B2),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7940_length__pos__if__in__set,axiom,
    ! [Xa: extended_enat,Xs: list_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),set_Extended_enat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3941691890525107288d_enat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_7941_length__pos__if__in__set,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),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_7942_length__pos__if__in__set,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_7943_length__pos__if__in__set,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),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_7944_length__pos__if__in__set,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xa,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_7945_length__pos__if__in__set,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),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_7946_length__pos__if__in__set,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),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_7947_card__length,axiom,
    ! [Xs: list_complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(set_complex2(Xs))),size_s3451745648224563538omplex(Xs)) ).

% card_length
tff(fact_7948_card__length,axiom,
    ! [Xs: list_set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(set_set_nat2(Xs))),size_s3254054031482475050et_nat(Xs)) ).

% card_length
tff(fact_7949_card__length,axiom,
    ! [Xs: list_Product_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(set_Product_unit2(Xs))),size_s245203480648594047t_unit(Xs)) ).

% card_length
tff(fact_7950_card__length,axiom,
    ! [Xs: list_list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(set_list_nat2(Xs))),size_s3023201423986296836st_nat(Xs)) ).

% card_length
tff(fact_7951_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs))),size_s6755466524823107622T_VEBT(Xs)) ).

% card_length
tff(fact_7952_card__length,axiom,
    ! [Xs: list_o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(set_o2(Xs))),size_size_list_o(Xs)) ).

% card_length
tff(fact_7953_card__length,axiom,
    ! [Xs: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(set_nat2(Xs))),size_size_list_nat(Xs)) ).

% card_length
tff(fact_7954_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% atLeast0_atMost_Suc
tff(fact_7955_atLeastAtMost__insertL,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( insert_nat(M2,set_or1269000886237332187st_nat(aa_nat_nat(suc,M2),Na)) = set_or1269000886237332187st_nat(M2,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_7956_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(suc,Na))
     => ( set_or1269000886237332187st_nat(M2,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(M2,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_7957_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( set_or1269000886237332187st_nat(M2,Na) = insert_nat(M2,set_or1269000886237332187st_nat(aa_nat_nat(suc,M2),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_7958_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or1269000886237332187st_nat(zero_zero_nat,Na))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_7959_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_7960_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_7961_complex__mod__minus__le__complex__mod,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xa))),aa_complex_real(real_V1022390504157884413omplex,Xa)) ).

% complex_mod_minus_le_complex_mod
tff(fact_7962_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A: 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),A))),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_7963_set__encode__insert,axiom,
    ! [A2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Na),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Na,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_7964_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7965_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7966_unset__bit__0,axiom,
    ! [A: int] : bit_se4203085406695923979it_int(zero_zero_nat,A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7967_signed__take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Na,A) = $ite(Na = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,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,Na),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7968_signed__take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Na),A) = $ite(Na = zero_zero_nat,uminus_uminus_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7969_set__union,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] : set_VEBT_VEBT2(union_VEBT_VEBT(Xs,Ys2)) = sup_su6272177626956685416T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(Ys2)) ).

% set_union
tff(fact_7970_set__union,axiom,
    ! [Xs: list_nat,Ys2: list_nat] : set_nat2(union_nat(Xs,Ys2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_nat2(Xs)),set_nat2(Ys2)) ).

% set_union
tff(fact_7971_set__union,axiom,
    ! [Xs: list_o,Ys2: list_o] : set_o2(union_o(Xs,Ys2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_o2(Xs)),set_o2(Ys2)) ).

% set_union
tff(fact_7972_set__union,axiom,
    ! [Xs: list_int,Ys2: list_int] : set_int2(union_int(Xs,Ys2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_int2(Xs)),set_int2(Ys2)) ).

% set_union
tff(fact_7973_round__unique,axiom,
    ! [Xa: real,Ya: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),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,Xa),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,Xa) = Ya ) ) ) ).

% round_unique
tff(fact_7974_round__unique,axiom,
    ! [Xa: rat,Ya: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,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(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xa) = Ya ) ) ) ).

% round_unique
tff(fact_7975_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_int(uminus_uminus_int(one_one_int)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7976_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_7977_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_7978_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7979_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_7980_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se4203085406695923979it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_7981_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se4203085406695923979it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_7982_signed__take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_ri631733984087533419it_int(Na),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_7983_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_7984_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_7985_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_int(zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_7986_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_7987_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_7988_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_7989_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_int(aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7990_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K)) = numera6620942414471956472nteger(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7991_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_numeral_dbl_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,K))) = uminus_uminus_int(neg_numeral_dbl_int(aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_7992_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_7993_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_7994_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_7995_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_7996_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_7997_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7998_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_7999_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_8000_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_int(one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_8001_dbl__simps_I3_J,axiom,
    neg_nu8804712462038260780nteger(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_8002_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_8003_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = uminus_uminus_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_8004_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se4203085406695923979it_int(Na,K)),K) ).

% unset_bit_less_eq
tff(fact_8005_set__encode__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B) )
        <=> ( A2 = B ) ) ) ) ).

% set_encode_eq
tff(fact_8006_dbl__def,axiom,
    ! [Xa: real] : aa_real_real(neg_numeral_dbl_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa) ).

% dbl_def
tff(fact_8007_dbl__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xa) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Xa) ).

% dbl_def
tff(fact_8008_dbl__def,axiom,
    ! [Xa: int] : neg_numeral_dbl_int(Xa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xa) ).

% dbl_def
tff(fact_8009_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na))
     => ( set_or1266510415728281911st_int(M2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na),set_or1266510415728281911st_int(M2,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8010_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).

% simp_from_to
tff(fact_8011_round__mono,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xa)),aa_rat_int(archim7778729529865785530nd_rat,Ya)) ) ).

% round_mono
tff(fact_8012_floor__le__round,axiom,
    ! [Xa: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xa)),aa_real_int(archim8280529875227126926d_real,Xa)) ).

% floor_le_round
tff(fact_8013_floor__le__round,axiom,
    ! [Xa: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xa)),aa_rat_int(archim7778729529865785530nd_rat,Xa)) ).

% floor_le_round
tff(fact_8014_ceiling__ge__round,axiom,
    ! [Xa: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xa)),aa_real_int(archim7802044766580827645g_real,Xa)) ).

% ceiling_ge_round
tff(fact_8015_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_8016_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X: int] :
              ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_8017_aset_I7_J,axiom,
    ! [D2: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa4) ) ) )
         => ( 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(aa_int_fun_int_int(plus_plus_int,X4),D2)) ) ) ) ).

% aset(7)
tff(fact_8018_aset_I5_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),A2)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa4) ) ) )
           => ( 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(aa_int_fun_int_int(plus_plus_int,X4),D2)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_8019_aset_I4_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),A2)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa4) ) ) )
           => ( ( X4 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D2) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_8020_aset_I3_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A2)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa4) ) ) )
           => ( ( X4 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D2) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_8021_bset_I7_J,axiom,
    ! [D2: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),B)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa4) ) ) )
           => ( 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(aa_int_fun_int_int(minus_minus_int,X4),D2)) ) ) ) ) ).

% bset(7)
tff(fact_8022_bset_I5_J,axiom,
    ! [D2: int,B: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa4) ) ) )
         => ( 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(aa_int_fun_int_int(minus_minus_int,X4),D2)),Ta) ) ) ) ).

% bset(5)
tff(fact_8023_bset_I4_J,axiom,
    ! [D2: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(Ta),B)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa4) ) ) )
           => ( ( X4 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D2) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_8024_bset_I3_J,axiom,
    ! [D2: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa4) ) ) )
           => ( ( X4 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D2) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_8025_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_8026_round__diff__minimal,axiom,
    ! [Z: real,M2: 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,M2)))) ).

% round_diff_minimal
tff(fact_8027_round__diff__minimal,axiom,
    ! [Z: rat,M2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,M2)))) ).

% round_diff_minimal
tff(fact_8028_bset_I6_J,axiom,
    ! [D2: int,B: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa4) ) ) )
         => ( 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(aa_int_fun_int_int(minus_minus_int,X4),D2)),Ta) ) ) ) ).

% bset(6)
tff(fact_8029_bset_I8_J,axiom,
    ! [D2: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa4) ) ) )
           => ( 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(aa_int_fun_int_int(minus_minus_int,X4),D2)) ) ) ) ) ).

% bset(8)
tff(fact_8030_aset_I6_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A2)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa4) ) ) )
           => ( 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(aa_int_fun_int_int(plus_plus_int,X4),D2)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_8031_aset_I8_J,axiom,
    ! [D2: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa4) ) ) )
         => ( 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(aa_int_fun_int_int(plus_plus_int,X4),D2)) ) ) ) ).

% aset(8)
tff(fact_8032_cpmi,axiom,
    ! [D2: 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),D2)
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P2,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P2,X2)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & aa_int_o(P2,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_8033_cppi,axiom,
    ! [D2: int,P: fun_int_o,P2: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P2,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D2))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P2,X2)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & aa_int_o(P2,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D2))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_8034_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_8035_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_8036_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_8037_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_8038_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),uminus_uminus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_8039_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_8040_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_8041_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_8042_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),uminus_uminus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_8043_of__int__round__le,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(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_8044_of__int__round__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_8045_of__int__round__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),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,Xa))) ).

% of_int_round_ge
tff(fact_8046_of__int__round__ge,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))) ).

% of_int_round_ge
tff(fact_8047_of__int__round__gt,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))) ).

% of_int_round_gt
tff(fact_8048_of__int__round__gt,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),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,Xa))) ).

% of_int_round_gt
tff(fact_8049_of__int__round__abs__le,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xa))),Xa))),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_8050_of__int__round__abs__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))),Xa))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_8051_round__unique_H,axiom,
    ! [Xa: rat,Na: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),aa_int_rat(ring_1_of_int_rat,Na)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,Xa) = Na ) ) ).

% round_unique'
tff(fact_8052_round__unique_H,axiom,
    ! [Xa: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_int_real(ring_1_of_int_real,Na)))),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,Xa) = Na ) ) ).

% round_unique'
tff(fact_8053_round__altdef,axiom,
    ! [Xa: real] :
      aa_real_int(archim8280529875227126926d_real,Xa) = $ite(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_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_int(archim7802044766580827645g_real,Xa),aa_real_int(archim6058952711729229775r_real,Xa)) ).

% round_altdef
tff(fact_8054_round__altdef,axiom,
    ! [Xa: rat] :
      aa_rat_int(archim7778729529865785530nd_rat,Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_int(archim2889992004027027881ng_rat,Xa),aa_rat_int(archim3151403230148437115or_rat,Xa)) ).

% round_altdef
tff(fact_8055_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_8056_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = aa_real_real(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,Xa)) ) ) ).

% log_base_10_eq1
tff(fact_8057_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(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)))),Na)),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,Na)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_8058_even__succ__mod__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8059_even__succ__mod__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8060_even__succ__mod__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_8061_even__succ__div__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_8062_even__succ__div__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_8063_even__succ__div__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_8064_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = aa_real_real(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,Xa)) ) ) ).

% log_base_10_eq2
tff(fact_8065_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),one_one_nat)
    <=> ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_8066_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_8067_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_8068_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_8069_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_8070_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_8071_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_8072_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_8073_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_8074_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_8075_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M2 = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_8076_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_8077_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_8078_tanh__real__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% tanh_real_less_iff
tff(fact_8079_tanh__real__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% tanh_real_le_iff
tff(fact_8080_semiring__norm_I73_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M2)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% semiring_norm(73)
tff(fact_8081_semiring__norm_I80_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M2)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% semiring_norm(80)
tff(fact_8082_dvd__mult__cancel__left,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),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,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_8083_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B2))
    <=> ( ( C2 = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_8084_dvd__mult__cancel__left,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))
    <=> ( ( C2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_8085_dvd__mult__cancel__right,axiom,
    ! [A: 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,A),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,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_8086_dvd__mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C2))
    <=> ( ( C2 = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_8087_dvd__mult__cancel__right,axiom,
    ! [A: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))
    <=> ( ( C2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_8088_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_8089_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_8090_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_8091_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_8092_dvd__imp__mod__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_8093_dvd__imp__mod__0,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_8094_binomial__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,zero_zero_nat)) = Na ).

% binomial_1
tff(fact_8095_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_8096_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_8097_binomial__n__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8098_semiring__norm_I72_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M2)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% semiring_norm(72)
tff(fact_8099_semiring__norm_I81_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M2)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% semiring_norm(81)
tff(fact_8100_semiring__norm_I70_J,axiom,
    ! [M2: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M2)),one) ).

% semiring_norm(70)
tff(fact_8101_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Na)) ).

% semiring_norm(77)
tff(fact_8102_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% tanh_real_pos_iff
tff(fact_8103_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_8104_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_8105_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_8106_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8107_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5851722552734809277nc_int(aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8108_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K)) = numera6620942414471956472nteger(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8109_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_8110_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_8111_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_8112_semiring__norm_I74_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M2)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% semiring_norm(74)
tff(fact_8113_semiring__norm_I79_J,axiom,
    ! [M2: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M2)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% semiring_norm(79)
tff(fact_8114_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8115_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8116_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8117_dbl__inc__simps_I3_J,axiom,
    neg_nu5851722552734809277nc_int(one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8118_dbl__inc__simps_I3_J,axiom,
    neg_nu5831290666863070958nteger(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8119_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_8120_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_8121_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_8122_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_8123_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_8124_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_8125_power__less__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_8126_power__less__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_8127_power__less__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_8128_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_8129_even__diff__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) ) ) ).

% even_diff_nat
tff(fact_8130_dbl__dec__simps_I4_J,axiom,
    neg_nu3811975205180677377ec_int(uminus_uminus_int(one_one_int)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8131_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_8132_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_8133_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8134_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_8135_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_8136_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_8137_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_8138_even__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_8139_even__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_8140_even__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),power_8256067586552552935nteger(A,Na))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_8141_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_8142_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_8143_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_8144_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_8145_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_8146_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: 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)),Na),one_one_Code_integer))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_8147_dvd__field__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
    <=> ( ( A = zero_zero_real )
       => ( B2 = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_8148_dvd__field__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2)
    <=> ( ( A = zero_zero_rat )
       => ( B2 = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_8149_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_8150_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_8151_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_8152_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_8153_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_8154_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_8155_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_8156_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_8157_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_8158_dvd__diff__nat,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) ) ) ).

% dvd_diff_nat
tff(fact_8159_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8160_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_8161_binomial__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R2)) ) ).

% binomial_le_pow
tff(fact_8162_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_8163_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_8164_pinf_I9_J,axiom,
    ! [D: real,S3: 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,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3)) ) ) ).

% pinf(9)
tff(fact_8165_pinf_I9_J,axiom,
    ! [D: rat,S3: 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,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3)) ) ) ).

% pinf(9)
tff(fact_8166_pinf_I9_J,axiom,
    ! [D: nat,S3: 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,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3)) ) ) ).

% pinf(9)
tff(fact_8167_pinf_I9_J,axiom,
    ! [D: int,S3: 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,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3)) ) ) ).

% pinf(9)
tff(fact_8168_pinf_I10_J,axiom,
    ! [D: real,S3: 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,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3)) ) ) ).

% pinf(10)
tff(fact_8169_pinf_I10_J,axiom,
    ! [D: rat,S3: 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,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3)) ) ) ).

% pinf(10)
tff(fact_8170_pinf_I10_J,axiom,
    ! [D: nat,S3: 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,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3)) ) ) ).

% pinf(10)
tff(fact_8171_pinf_I10_J,axiom,
    ! [D: int,S3: 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,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3)) ) ) ).

% pinf(10)
tff(fact_8172_minf_I9_J,axiom,
    ! [D: real,S3: 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,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3)) ) ) ).

% minf(9)
tff(fact_8173_minf_I9_J,axiom,
    ! [D: rat,S3: 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,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3)) ) ) ).

% minf(9)
tff(fact_8174_minf_I9_J,axiom,
    ! [D: nat,S3: 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,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3)) ) ) ).

% minf(9)
tff(fact_8175_minf_I9_J,axiom,
    ! [D: int,S3: 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,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3)) ) ) ).

% minf(9)
tff(fact_8176_minf_I10_J,axiom,
    ! [D: real,S3: 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,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S3)) ) ) ).

% minf(10)
tff(fact_8177_minf_I10_J,axiom,
    ! [D: rat,S3: 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,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S3)) ) ) ).

% minf(10)
tff(fact_8178_minf_I10_J,axiom,
    ! [D: nat,S3: 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,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S3)) ) ) ).

% minf(10)
tff(fact_8179_minf_I10_J,axiom,
    ! [D: int,S3: 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,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S3)) ) ) ).

% minf(10)
tff(fact_8180_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B2) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8181_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8182_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8183_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8184_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B2),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_8185_num_Oexhaust,axiom,
    ! [Ya: num] :
      ( ( Ya != one )
     => ( ! [X24: num] : Ya != aa_num_num(bit0,X24)
       => ~ ! [X32: num] : Ya != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_8186_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_8187_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_8188_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_8189_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_8190_mod__0__imp__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ).

% mod_0_imp_dvd
tff(fact_8191_mod__0__imp__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ).

% mod_0_imp_dvd
tff(fact_8192_le__imp__power__dvd,axiom,
    ! [M2: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8193_le__imp__power__dvd,axiom,
    ! [M2: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8194_le__imp__power__dvd,axiom,
    ! [M2: nat,Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8195_le__imp__power__dvd,axiom,
    ! [M2: nat,Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_8196_power__le__dvd,axiom,
    ! [A: int,Na: nat,B2: int,M2: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M2)),B2) ) ) ).

% power_le_dvd
tff(fact_8197_power__le__dvd,axiom,
    ! [A: nat,Na: nat,B2: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),B2) ) ) ).

% power_le_dvd
tff(fact_8198_power__le__dvd,axiom,
    ! [A: real,Na: nat,B2: real,M2: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),B2) ) ) ).

% power_le_dvd
tff(fact_8199_power__le__dvd,axiom,
    ! [A: complex,Na: nat,B2: complex,M2: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Na)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M2)),B2) ) ) ).

% power_le_dvd
tff(fact_8200_dvd__power__le,axiom,
    ! [Xa: int,Ya: int,Na: nat,M2: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),Na)),aa_nat_int(power_power_int(Ya),M2)) ) ) ).

% dvd_power_le
tff(fact_8201_dvd__power__le,axiom,
    ! [Xa: nat,Ya: nat,Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),M2)) ) ) ).

% dvd_power_le
tff(fact_8202_dvd__power__le,axiom,
    ! [Xa: real,Ya: real,Na: nat,M2: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),M2)) ) ) ).

% dvd_power_le
tff(fact_8203_dvd__power__le,axiom,
    ! [Xa: complex,Ya: complex,Na: nat,M2: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xa),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xa),Na)),aa_nat_complex(power_power_complex(Ya),M2)) ) ) ).

% dvd_power_le
tff(fact_8204_dvd__pos__nat,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2) ) ) ).

% dvd_pos_nat
tff(fact_8205_nat__dvd__not__less,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M2) ) ) ).

% nat_dvd_not_less
tff(fact_8206_dvd__minus__self,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na) ) ) ).

% dvd_minus_self
tff(fact_8207_zdvd__antisym__nonneg,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M2),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M2)
           => ( M2 = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_8208_dvd__diffD,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M2) ) ) ) ).

% dvd_diffD
tff(fact_8209_dvd__diffD1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_8210_less__eq__dvd__minus,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) ) ) ).

% less_eq_dvd_minus
tff(fact_8211_zdvd__not__zless,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M2),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M2) ) ) ).

% zdvd_not_zless
tff(fact_8212_fact__dvd,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,M2)) ) ).

% fact_dvd
tff(fact_8213_fact__dvd,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,M2)) ) ).

% fact_dvd
tff(fact_8214_fact__dvd,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,M2)) ) ).

% fact_dvd
tff(fact_8215_dvd__Gcd__fin__iff,axiom,
    ! [A2: set_nat,B2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),semiri4258706085729940814in_nat(A2))
      <=> ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A2)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),X) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_8216_dvd__Gcd__fin__iff,axiom,
    ! [A2: set_int,B2: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),semiri4256215615220890538in_int(A2))
      <=> ! [X: int] :
            ( aa_set_int_o(member_int(X),A2)
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),X) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_8217_Gcd__fin__greatest,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [B4: nat] :
            ( aa_set_nat_o(member_nat(B4),A2)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B4) )
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),semiri4258706085729940814in_nat(A2)) ) ) ).

% Gcd_fin_greatest
tff(fact_8218_Gcd__fin__greatest,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [B4: int] :
            ( aa_set_int_o(member_int(B4),A2)
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B4) )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),semiri4256215615220890538in_int(A2)) ) ) ).

% Gcd_fin_greatest
tff(fact_8219_tanh__real__lt__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),one_one_real) ).

% tanh_real_lt_1
tff(fact_8220_nat__dvd__iff,axiom,
    ! [Z: int,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M2)
    <=> $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,M2)),M2 = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_8221_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ).

% zero_less_binomial
tff(fact_8222_choose__mult,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),M2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K))) ) ) ) ).

% choose_mult
tff(fact_8223_unit__dvdE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C4: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C4) ) ) ).

% unit_dvdE
tff(fact_8224_unit__dvdE,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C4: int] : B2 != aa_int_int(aa_int_fun_int_int(times_times_int,A),C4) ) ) ).

% unit_dvdE
tff(fact_8225_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X))
    <=> ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),zero_zero_real))
          & aa_real_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_8226_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X))
    <=> ? [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),zero_zero_rat))
          & aa_rat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_8227_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X))
    <=> ? [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat))
          & aa_nat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_8228_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X))
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),zero_zero_int))
          & aa_int_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_8229_dvd__div__div__eq__mult,axiom,
    ! [A: int,C2: int,B2: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C2) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8230_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C2: nat,B2: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),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,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_8231_dvd__div__iff__mult,axiom,
    ! [C2: int,B2: int,A: 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,A),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(aa_int_fun_int_int(times_times_int,A),C2)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8232_dvd__div__iff__mult,axiom,
    ! [C2: nat,B2: nat,A: 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,A),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,A),C2)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_8233_div__dvd__iff__mult,axiom,
    ! [B2: int,A: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C2)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8234_div__dvd__iff__mult,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C2)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_8235_dvd__div__eq__mult,axiom,
    ! [A: int,B2: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) = C2 )
        <=> ( B2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8236_dvd__div__eq__mult,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A) = C2 )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_8237_unit__div__eq__0__iff,axiom,
    ! [B2: int,A: 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,A),B2) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_8238_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A: 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,A),B2) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_8239_numeral__Bit1,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit1,Na)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),numera6690914467698888265omplex(Na))),one_one_complex) ).

% numeral_Bit1
tff(fact_8240_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na))),one_one_rat) ).

% numeral_Bit1
tff(fact_8241_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na))),one_one_real) ).

% numeral_Bit1
tff(fact_8242_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na))),one_one_nat) ).

% numeral_Bit1
tff(fact_8243_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na))),one_one_int) ).

% numeral_Bit1
tff(fact_8244_numeral__Bit1,axiom,
    ! [Na: num] : numera1916890842035813515d_enat(aa_num_num(bit1,Na)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_8245_numeral__Bit1,axiom,
    ! [Na: num] : numera6620942414471956472nteger(aa_num_num(bit1,Na)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Na)),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_8246_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_8247_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A: 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,A),B2) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_8248_is__unit__power__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_8249_is__unit__power__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_8250_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ).

% eval_nat_numeral(3)
tff(fact_8251_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_8252_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_8253_dvd__mult__cancel,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na) ) ) ).

% dvd_mult_cancel
tff(fact_8254_bezout__add__strong__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X2: nat,Y: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),B2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = 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_8255_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_8256_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M2) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_8257_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_8258_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M2: nat,Q5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q5) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q5) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_8259_dvd__fact,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% dvd_fact
tff(fact_8260_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_8261_tanh__real__gt__neg1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xa)) ).

% tanh_real_gt_neg1
tff(fact_8262_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_8263_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_8264_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_8265_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B2: int] :
      ( ( A != 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_8266_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_8267_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_8268_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_8269_is__unitE,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B4),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B4) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C2),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8270_is__unitE,axiom,
    ! [A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B4),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_8271_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_8272_dvd__power__iff,axiom,
    ! [Xa: int,M2: nat,Na: nat] :
      ( ( Xa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),M2)),aa_nat_int(power_power_int(Xa),Na))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ) ).

% dvd_power_iff
tff(fact_8273_dvd__power__iff,axiom,
    ! [Xa: nat,M2: nat,Na: nat] :
      ( ( Xa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ) ).

% dvd_power_iff
tff(fact_8274_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_8275_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q5: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q5))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_8276_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q5: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q5))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_8277_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q5: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q5))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_8278_dvd__power,axiom,
    ! [Na: nat,Xa: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),aa_nat_rat(power_power_rat(Xa),Na)) ) ).

% dvd_power
tff(fact_8279_dvd__power,axiom,
    ! [Na: nat,Xa: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),aa_nat_int(power_power_int(Xa),Na)) ) ).

% dvd_power
tff(fact_8280_dvd__power,axiom,
    ! [Na: nat,Xa: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na)) ) ).

% dvd_power
tff(fact_8281_dvd__power,axiom,
    ! [Na: nat,Xa: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) ) ).

% dvd_power
tff(fact_8282_dvd__power,axiom,
    ! [Na: nat,Xa: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xa = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xa),aa_nat_complex(power_power_complex(Xa),Na)) ) ).

% dvd_power
tff(fact_8283_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_8284_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na) ).

% Suc3_eq_add_3
tff(fact_8285_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% choose_dvd
tff(fact_8286_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% choose_dvd
tff(fact_8287_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% choose_dvd
tff(fact_8288_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% choose_dvd
tff(fact_8289_dvd__mult__cancel1,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)),M2)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_8290_dvd__mult__cancel2,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M2)),M2)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_8291_dvd__minus__add,axiom,
    ! [Q5: nat,Na: nat,R2: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q5),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q5),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M2))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Q5))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M2)),Q5))) ) ) ) ).

% dvd_minus_add
tff(fact_8292_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% power_dvd_imp_le
tff(fact_8293_mod__nat__eqI,axiom,
    ! [R2: nat,Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),M2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_8294_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_8295_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8296_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8297_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Na)) ).

% binomial_maximum'
tff(fact_8298_binomial__mono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K7)
     => ( 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))),K7)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K7)) ) ) ).

% binomial_mono
tff(fact_8299_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8300_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K7)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K7),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K7)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_8301_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% binomial_le_pow2
tff(fact_8302_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8303_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8304_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_8305_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_8306_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_8307_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_8308_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8309_cong__exp__iff__simps_I7_J,axiom,
    ! [Q5: num,Na: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q5))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q5))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q5)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8310_cong__exp__iff__simps_I7_J,axiom,
    ! [Q5: num,Na: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q5))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q5))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q5)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8311_cong__exp__iff__simps_I7_J,axiom,
    ! [Q5: num,Na: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q5))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q5))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q5)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8312_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q5: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M2)),numera6620942414471956472nteger(aa_num_num(bit0,Q5))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q5))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(Q5)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8313_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q5: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q5))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q5))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Q5)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8314_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q5: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q5))) = 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,Q5))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Q5)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8315_power__mono__odd,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na)) ) ) ).

% power_mono_odd
tff(fact_8316_power__mono__odd,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B2),Na)) ) ) ).

% power_mono_odd
tff(fact_8317_power__mono__odd,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na)) ) ) ).

% power_mono_odd
tff(fact_8318_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_8319_card__3__iff,axiom,
    ! [S: set_complex] :
      ( ( finite_card_complex(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: complex,Y4: complex,Z5: complex] :
          ( ( S = insert_complex(X,insert_complex(Y4,insert_complex(Z5,bot_bot_set_complex))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8320_card__3__iff,axiom,
    ! [S: set_Product_unit] :
      ( ( finite410649719033368117t_unit(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: product_unit,Y4: product_unit,Z5: product_unit] :
          ( ( S = insert_Product_unit(X,insert_Product_unit(Y4,insert_Product_unit(Z5,bot_bo3957492148770167129t_unit))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8321_card__3__iff,axiom,
    ! [S: set_list_nat] :
      ( ( finite_card_list_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: list_nat,Y4: list_nat,Z5: list_nat] :
          ( ( S = insert_list_nat(X,insert_list_nat(Y4,insert_list_nat(Z5,bot_bot_set_list_nat))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8322_card__3__iff,axiom,
    ! [S: set_nat] :
      ( ( finite_card_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: nat,Y4: nat,Z5: nat] :
          ( ( S = insert_nat(X,insert_nat(Y4,insert_nat(Z5,bot_bot_set_nat))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8323_card__3__iff,axiom,
    ! [S: set_int] :
      ( ( finite_card_int(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: int,Y4: int,Z5: int] :
          ( ( S = insert_int(X,insert_int(Y4,insert_int(Z5,bot_bot_set_int))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8324_card__3__iff,axiom,
    ! [S: set_o] :
      ( ( finite_card_o(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: $o,Y4: $o,Z5: $o] :
          ( ( S = insert_o((X),insert_o((Y4),insert_o((Z5),bot_bot_set_o))) )
          & ~ ( (X)
            <=> (Y4) )
          & ~ ( (Y4)
            <=> (Z5) )
          & ~ ( (X)
            <=> (Z5) ) ) ) ).

% card_3_iff
tff(fact_8325_card__3__iff,axiom,
    ! [S: set_set_nat] :
      ( ( finite_card_set_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: set_nat,Y4: set_nat,Z5: set_nat] :
          ( ( S = insert_set_nat(X,insert_set_nat(Y4,insert_set_nat(Z5,bot_bot_set_set_nat))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8326_card__3__iff,axiom,
    ! [S: set_real] :
      ( ( finite_card_real(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: real,Y4: real,Z5: real] :
          ( ( S = insert_real(X,insert_real(Y4,insert_real(Z5,bot_bot_set_real))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8327_card__3__iff,axiom,
    ! [S: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X: extended_enat,Y4: extended_enat,Z5: extended_enat] :
          ( ( S = insert_Extended_enat(X,insert_Extended_enat(Y4,insert_Extended_enat(Z5,bot_bo7653980558646680370d_enat))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_8328_odd__card__imp__not__empty,axiom,
    ! [A2: set_complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_complex(A2))
     => ( A2 != bot_bot_set_complex ) ) ).

% odd_card_imp_not_empty
tff(fact_8329_odd__card__imp__not__empty,axiom,
    ! [A2: set_Product_unit] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite410649719033368117t_unit(A2))
     => ( A2 != bot_bo3957492148770167129t_unit ) ) ).

% odd_card_imp_not_empty
tff(fact_8330_odd__card__imp__not__empty,axiom,
    ! [A2: set_list_nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_list_nat(A2))
     => ( A2 != bot_bot_set_list_nat ) ) ).

% odd_card_imp_not_empty
tff(fact_8331_odd__card__imp__not__empty,axiom,
    ! [A2: set_nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_nat(A2))
     => ( A2 != bot_bot_set_nat ) ) ).

% odd_card_imp_not_empty
tff(fact_8332_odd__card__imp__not__empty,axiom,
    ! [A2: set_int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_int(A2))
     => ( A2 != bot_bot_set_int ) ) ).

% odd_card_imp_not_empty
tff(fact_8333_odd__card__imp__not__empty,axiom,
    ! [A2: set_o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(A2))
     => ( A2 != bot_bot_set_o ) ) ).

% odd_card_imp_not_empty
tff(fact_8334_odd__card__imp__not__empty,axiom,
    ! [A2: set_set_nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_set_nat(A2))
     => ( A2 != bot_bot_set_set_nat ) ) ).

% odd_card_imp_not_empty
tff(fact_8335_odd__card__imp__not__empty,axiom,
    ! [A2: set_real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_real(A2))
     => ( A2 != bot_bot_set_real ) ) ).

% odd_card_imp_not_empty
tff(fact_8336_odd__card__imp__not__empty,axiom,
    ! [A2: set_Extended_enat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite121521170596916366d_enat(A2))
     => ( A2 != bot_bo7653980558646680370d_enat ) ) ).

% odd_card_imp_not_empty
tff(fact_8337_dvd__power__iff__le,axiom,
    ! [K: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% dvd_power_iff_le
tff(fact_8338_even__unset__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_8339_even__unset__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M2,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_8340_even__unset__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se4203085406695923979it_int(M2,A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_8341_even__set__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_8342_even__set__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M2,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_8343_even__set__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se7879613467334960850it_int(M2,A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_8344_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_8345_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_8346_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K7)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K7),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K7)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8347_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K7)
     => ( 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))),K7)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K7)) ) ) ).

% binomial_strict_mono
tff(fact_8348_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8349_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_8350_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Na)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_8351_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_8352_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_8353_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Na)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_8354_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_8355_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,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))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_8356_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_8357_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_8358_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_8359_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_8360_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_8361_zero__le__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8362_zero__le__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8363_zero__le__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_8364_zero__le__odd__power,axiom,
    ! [Na: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_8365_zero__le__odd__power,axiom,
    ! [Na: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_8366_zero__le__odd__power,axiom,
    ! [Na: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_8367_zero__le__even__power,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_even_power
tff(fact_8368_zero__le__even__power,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_even_power
tff(fact_8369_zero__le__even__power,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_even_power
tff(fact_8370_power__mono__even,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B2))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A,Na)),power_8256067586552552935nteger(B2,Na)) ) ) ).

% power_mono_even
tff(fact_8371_power__mono__even,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Na)) ) ) ).

% power_mono_even
tff(fact_8372_power__mono__even,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B2),Na)) ) ) ).

% power_mono_even
tff(fact_8373_power__mono__even,axiom,
    ! [Na: nat,A: 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))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B2),Na)) ) ) ).

% power_mono_even
tff(fact_8374_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),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,M2),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,M2),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,M2),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_8375_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_8376_zero__less__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8377_zero__less__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8378_zero__less__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_8379_even__mask__div__iff_H,axiom,
    ! [M2: nat,Na: 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)),M2),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% even_mask_div_iff'
tff(fact_8380_even__mask__div__iff_H,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% even_mask_div_iff'
tff(fact_8381_even__mask__div__iff_H,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% even_mask_div_iff'
tff(fact_8382_power__le__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8383_power__le__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8384_power__le__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_8385_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_8386_even__mask__div__iff,axiom,
    ! [M2: nat,Na: 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)),M2),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)))
    <=> ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% even_mask_div_iff
tff(fact_8387_even__mask__div__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% even_mask_div_iff
tff(fact_8388_even__mask__div__iff,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ) ).

% even_mask_div_iff
tff(fact_8389_Bernoulli__inequality__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_8390_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M2: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(times_3573771949741848930nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
        | ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8391_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8392_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_8393_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_8394_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),one_one_nat)),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_8395_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_8396_take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Na,A) = $ite(Na = zero_zero_nat,zero_z3403309356797280102nteger,plus_p5714425477246183910nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_8397_take__bit__rec,axiom,
    ! [Na: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_8398_take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),A) = $ite(Na = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_8399_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : aa_num_nat(size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_8400_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Na),nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Z)) = insert_nat(Na,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_8401_take__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_8402_take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_8403_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_8404_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_8405_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8406_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_8407_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_8408_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = zero_zero_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_8409_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),one_one_int) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_8410_even__take__bit__eq,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Na,A))
    <=> ( ( Na = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_8411_even__take__bit__eq,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))
    <=> ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_8412_even__take__bit__eq,axiom,
    ! [Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(Na),A))
    <=> ( ( Na = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_8413_set__decode__0,axiom,
    ! [Xa: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(Xa))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa) ) ).

% set_decode_0
tff(fact_8414_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_8415_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_8416_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_8417_dvd__antisym,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M2)
       => ( M2 = Na ) ) ) ).

% dvd_antisym
tff(fact_8418_take__bit__tightened,axiom,
    ! [Na: nat,A: nat,B2: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),B2) ) ) ) ).

% take_bit_tightened
tff(fact_8419_take__bit__tightened,axiom,
    ! [Na: nat,A: int,B2: int,M2: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = aa_int_int(bit_se2923211474154528505it_int(Na),B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
       => ( aa_int_int(bit_se2923211474154528505it_int(M2),A) = aa_int_int(bit_se2923211474154528505it_int(M2),B2) ) ) ) ).

% take_bit_tightened
tff(fact_8420_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M2)),M2) ).

% take_bit_nat_less_eq_self
tff(fact_8421_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,Na: nat,Q5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),Q5)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Q5)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8422_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_8423_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_8424_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M2),K)),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8425_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_8426_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% take_bit_nonnegative
tff(fact_8427_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_8428_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_8429_signed__take__bit__take__bit,axiom,
    ! [M2: nat,Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(M2),aa_int_int(bit_se2923211474154528505it_int(Na),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),bit_se2923211474154528505it_int(Na),bit_ri631733984087533419it_int(M2)),A) ).

% signed_take_bit_take_bit
tff(fact_8430_finite__set__decode,axiom,
    ! [Na: nat] : aa_set_nat_o(finite_finite_nat,nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_8431_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M2: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_unset_bit_eq
tff(fact_8432_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M2: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),bit_se4203085406695923979it_int(M2,A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_int_int(bit_se2923211474154528505it_int(Na),A),bit_se4203085406695923979it_int(M2,aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_unset_bit_eq
tff(fact_8433_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M2: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_set_bit_eq
tff(fact_8434_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M2: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),bit_se7879613467334960850it_int(M2,A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_int_int(bit_se2923211474154528505it_int(Na),A),bit_se7879613467334960850it_int(M2,aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_set_bit_eq
tff(fact_8435_take__bit__signed__take__bit,axiom,
    ! [M2: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(suc,Na))
     => ( aa_int_int(bit_se2923211474154528505it_int(M2),aa_int_int(bit_ri631733984087533419it_int(Na),A)) = aa_int_int(bit_se2923211474154528505it_int(M2),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_8436_subset__decode__imp__le,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,nat_set_decode(M2)),nat_set_decode(Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% subset_decode_imp_le
tff(fact_8437_take__bit__nat__eq__self,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M2) = M2 ) ) ).

% take_bit_nat_eq_self
tff(fact_8438_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_8439_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M2) = M2 )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_8440_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_less_exp
tff(fact_8441_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xa: fun_nat_fun_nat_nat,Xaa: nat,Xb3: nat,Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xa,Xaa),Xb3),Xc) = Ya )
     => ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb3),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xb3),aa_nat_nat(aa_nat_fun_nat_nat(Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_8442_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B2: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B2),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A),Acc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_8443_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_8444_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Na,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_8445_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_8446_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_8447_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M2)),M2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M2) ) ).

% take_bit_nat_less_self_iff
tff(fact_8448_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_8449_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8450_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_8451_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8452_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_8453_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_8454_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Na,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_8455_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_8456_stable__imp__take__bit__eq,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_8457_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_se2923211474154528505it_int(Na),uminus_uminus_int(K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_8458_modulo__int__unfold,axiom,
    ! [K: int,M2: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( sgn_sgn_int(L) = zero_zero_int )
        | ( sgn_sgn_int(K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),aa_nat_int(semiri1314217659103216013at_int,M2)),
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na))),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M2))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Na))))) ) ).

% modulo_int_unfold
tff(fact_8459_divide__int__unfold,axiom,
    ! [K: int,M2: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(K)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( sgn_sgn_int(L) = zero_zero_int )
        | ( sgn_sgn_int(K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        zero_zero_int,
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na)),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,M2),Na)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M2)))))) ) ).

% divide_int_unfold
tff(fact_8460_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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),Xa)
         => ( 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,Xa),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_8461_even__flip__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_8462_even__flip__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M2,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_8463_even__flip__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se2159334234014336723it_int(M2,A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_8464_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : modulo364778990260209775nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_8465_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_8466_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_8467_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se2159334234014336723it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_8468_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2159334234014336723it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_8469_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),aa_o_rat(zero_n2052037380579107095ol_rat,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_8470_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_8471_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_8472_of__bool__eq_I1_J,axiom,
    aa_o_real(zero_n3304061248610475627l_real,$false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_8473_of__bool__eq_I1_J,axiom,
    aa_o_rat(zero_n2052037380579107095ol_rat,$false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_8474_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_8475_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_8476_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_real(zero_n3304061248610475627l_real,(P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_8477_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_rat(zero_n2052037380579107095ol_rat,(P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_8478_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_8479_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = zero_zero_int )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_8480_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_8481_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),aa_o_rat(zero_n2052037380579107095ol_rat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_8482_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_8483_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_8484_real__sqrt__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ).

% real_sqrt_less_iff
tff(fact_8485_real__sqrt__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ).

% real_sqrt_le_iff
tff(fact_8486_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_8487_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_8488_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_8489_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_8490_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_8491_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_8492_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_8493_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_8494_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_8495_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8496_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_8497_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_8498_real__sqrt__lt__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_8499_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_8500_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_8501_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_8502_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% of_nat_of_bool
tff(fact_8503_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_8504_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_8505_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(sgn_sgn_Code_integer(A),sgn_sgn_Code_integer(A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_8506_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% sgn_mult_self_eq
tff(fact_8507_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = aa_o_rat(zero_n2052037380579107095ol_rat,A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_8508_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,sgn_sgn_int(A)),sgn_sgn_int(A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_8509_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).

% sgn_abs
tff(fact_8510_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(A != zero_zero_complex) ).

% sgn_abs
tff(fact_8511_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% sgn_abs
tff(fact_8512_sgn__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_o_rat(zero_n2052037380579107095ol_rat,A != zero_zero_rat) ).

% sgn_abs
tff(fact_8513_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,sgn_sgn_int(A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_abs
tff(fact_8514_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(abs_abs_Code_integer(A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8515_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(A != zero_zero_complex) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8516_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8517_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(abs_abs_rat,A)) = aa_o_rat(zero_n2052037380579107095ol_rat,A != zero_zero_rat) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8518_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : sgn_sgn_int(aa_int_int(abs_abs_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8519_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_o_nat(zero_n2687167440665602831ol_nat,Na != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_8520_take__bit__of__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_8521_take__bit__of__1,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_8522_sgn__of__nat,axiom,
    ! [Na: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_8523_sgn__of__nat,axiom,
    ! [Na: nat] : sgn_sgn_Code_integer(semiri4939895301339042750nteger(Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_8524_sgn__of__nat,axiom,
    ! [Na: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_o_real(zero_n3304061248610475627l_real,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_8525_sgn__of__nat,axiom,
    ! [Na: nat] : sgn_sgn_int(aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_8526_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_Suc_0
tff(fact_8527_of__bool__half__eq__0,axiom,
    ! [B2: $o] : divide6298287555418463151nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B2)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_8528_of__bool__half__eq__0,axiom,
    ! [B2: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_8529_of__bool__half__eq__0,axiom,
    ! [B2: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_8530_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_8531_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_8532_bits__1__div__exp,axiom,
    ! [Na: nat] : divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_8533_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_8534_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_8535_one__div__2__pow__eq,axiom,
    ! [Na: nat] : divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_8536_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_8537_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_8538_take__bit__of__exp,axiom,
    ! [M2: nat,Na: nat] : bit_se1745604003318907178nteger(M2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = times_3573771949741848930nteger(aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) ).

% take_bit_of_exp
tff(fact_8539_take__bit__of__exp,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_8540_take__bit__of__exp,axiom,
    ! [M2: nat,Na: nat] : aa_int_int(bit_se2923211474154528505it_int(M2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_8541_take__bit__of__2,axiom,
    ! [Na: nat] : bit_se1745604003318907178nteger(Na,numera6620942414471956472nteger(aa_num_num(bit0,one))) = times_3573771949741848930nteger(aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8542_take__bit__of__2,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8543_take__bit__of__2,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_8544_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_8545_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_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))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_8546_flip__bit__0,axiom,
    ! [A: int] : bit_se2159334234014336723it_int(zero_zero_nat,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_8547_real__sqrt__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Ya)) ) ).

% real_sqrt_less_mono
tff(fact_8548_real__sqrt__le__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Ya)) ) ).

% real_sqrt_le_mono
tff(fact_8549_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_8550_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sqrt,Xa) = zero_zero_real )
       => ( Xa = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_8551_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_8552_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P))) ).

% zero_less_eq_of_bool
tff(fact_8553_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(P))) ).

% zero_less_eq_of_bool
tff(fact_8554_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_8555_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% zero_less_eq_of_bool
tff(fact_8556_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_one
tff(fact_8557_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_8558_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_8559_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_8560_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_8561_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n1201886186963655149omplex((P3)) = $ite((P3),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_8562_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_real(zero_n3304061248610475627l_real,(P3)) = $ite((P3),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_8563_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_rat(zero_n2052037380579107095ol_rat,(P3)) = $ite((P3),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_8564_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P3)) = $ite((P3),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_8565_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P3)) = $ite((P3),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_8566_split__of__bool,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ( ( (P3)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P3)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_8567_split__of__bool,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,aa_o_real(zero_n3304061248610475627l_real,(P3)))
    <=> ( ( (P3)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P3)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_8568_split__of__bool,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P3)))
    <=> ( ( (P3)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P3)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_8569_split__of__bool,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ( ( (P3)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P3)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_8570_split__of__bool,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ( ( (P3)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P3)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_8571_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P3)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_8572_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,aa_o_real(zero_n3304061248610475627l_real,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P3)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_8573_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P3)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_8574_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P3)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_8575_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P3)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_8576_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_real_real(sqrt,Xa)) = aa_real_real(sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_8577_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Ya))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_8578_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)))) ).

% le_real_sqrt_sumsq
tff(fact_8579_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_8580_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xa)),Xa) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_8581_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M2: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_flip_bit_eq
tff(fact_8582_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M2: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),bit_se2159334234014336723it_int(M2,A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2),aa_int_int(bit_se2923211474154528505it_int(Na),A),bit_se2159334234014336723it_int(M2,aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_flip_bit_eq
tff(fact_8583_real__less__rsqrt,axiom,
    ! [Xa: 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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(sqrt,Ya)) ) ).

% real_less_rsqrt
tff(fact_8584_real__le__rsqrt,axiom,
    ! [Xa: 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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,Ya)) ) ).

% real_le_rsqrt
tff(fact_8585_sqrt__le__D,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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_8586_real__sqrt__unique,axiom,
    ! [Ya: real,Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_real(sqrt,Xa) = Ya ) ) ) ).

% real_sqrt_unique
tff(fact_8587_real__le__lsqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),Ya) ) ) ) ).

% real_le_lsqrt
tff(fact_8588_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_8589_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_8590_real__sqrt__sum__squares__ge2,axiom,
    ! [Ya: real,Xa: 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,Xa),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_8591_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C2: real,B2: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),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),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_8592_sqrt__ge__absD,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya) ) ).

% sqrt_ge_absD
tff(fact_8593_real__less__lsqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)),Ya) ) ) ) ).

% real_less_lsqrt
tff(fact_8594_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa),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,Xa),Ya)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8595_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8596_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: 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,Xa),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,Xa)),aa_real_real(abs_abs_real,Ya))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8597_real__sqrt__ge__abs2,axiom,
    ! [Ya: real,Xa: 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,Xa),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_8598_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_8599_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8600_exp__mod__exp,axiom,
    ! [M2: nat,Na: nat] : modulo364778990260209775nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = times_3573771949741848930nteger(aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2)) ).

% exp_mod_exp
tff(fact_8601_exp__mod__exp,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_8602_exp__mod__exp,axiom,
    ! [M2: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_8603_arsinh__real__aux,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8604_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: 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,Xa),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_8605_real__sqrt__power__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8606_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8607_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),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,Xa) ) ) ).

% powr_half_sqrt
tff(fact_8608_cos__x__y__le__one,axiom,
    ! [Xa: 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,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_8609_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),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,Xa),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_8610_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_real(arcosh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8611_exp__div__exp__eq,axiom,
    ! [M2: nat,Na: nat] :
      divide6298287555418463151nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = times_3573771949741848930nteger(aa_o_Code_integer(zero_n356916108424825756nteger,
          ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M2) != zero_z3403309356797280102nteger )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) )),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na))) ).

% exp_div_exp_eq
tff(fact_8612_exp__div__exp__eq,axiom,
    ! [M2: nat,Na: 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))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na))) ).

% exp_div_exp_eq
tff(fact_8613_exp__div__exp__eq,axiom,
    ! [M2: nat,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2) ))),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,M2),Na))) ).

% exp_div_exp_eq
tff(fact_8614_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8615_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na)))),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))),Na))) ).

% Suc_0_xor_eq
tff(fact_8616_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na)))),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))),Na))) ).

% xor_Suc_0_eq
tff(fact_8617_cosh__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cosh_real,Xa) = zero_zero_real )
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_8618_cosh__zero__iff,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(cosh_complex,Xa) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_8619_xor__nat__unfold,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),Na) = $ite(
        M2 = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M2,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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_8620_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_8621_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_8622_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_8623_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_8624_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_8625_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_8626_bit_Oxor__self,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Xa) = zero_zero_int ).

% bit.xor_self
tff(fact_8627_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_8628_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_8629_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_8630_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_8631_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% xor_nat_numerals(3)
tff(fact_8632_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa)) ).

% xor_nat_numerals(4)
tff(fact_8633_cosh__real__pos,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_pos
tff(fact_8634_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_8635_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_8636_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_nonneg
tff(fact_8637_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_8638_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_ge_1
tff(fact_8639_sinh__less__cosh__real,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),aa_real_real(cosh_real,Xa)) ).

% sinh_less_cosh_real
tff(fact_8640_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),aa_real_real(cosh_real,Xa)) ).

% sinh_le_cosh_real
tff(fact_8641_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8642_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8643_cosh__real__strict__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Ya)) ) ) ).

% cosh_real_strict_mono
tff(fact_8644_tanh__add,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(cosh_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Ya) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Ya)))) ) ) ) ).

% tanh_add
tff(fact_8645_tanh__add,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_complex(cosh_complex,Xa) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Ya) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(Xa),Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,Xa)),aa_complex_complex(tanh_complex,Ya))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,Xa)),aa_complex_complex(tanh_complex,Ya)))) ) ) ) ).

% tanh_add
tff(fact_8646_drop__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_se3928097537394005634nteger(Na,A) = $ite(Na = zero_zero_nat,A,bit_se3928097537394005634nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_8647_drop__bit__rec,axiom,
    ! [Na: nat,A: int] :
      bit_se8568078237143864401it_int(Na,A) = $ite(Na = zero_zero_nat,A,bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_8648_drop__bit__rec,axiom,
    ! [Na: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),A) = $ite(Na = zero_zero_nat,A,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% drop_bit_rec
tff(fact_8649_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na))) ).

% Suc_0_or_eq
tff(fact_8650_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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))),Na))) ).

% or_Suc_0_eq
tff(fact_8651_and__nat__unfold,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),Na) = $ite(
        ( ( M2 = zero_zero_nat )
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_8652_is__empty__set,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( is_empty_VEBT_VEBT(set_VEBT_VEBT2(Xs))
    <=> aa_list_VEBT_VEBT_o(null_VEBT_VEBT,Xs) ) ).

% is_empty_set
tff(fact_8653_is__empty__set,axiom,
    ! [Xs: list_nat] :
      ( aa_set_nat_o(is_empty_nat,set_nat2(Xs))
    <=> aa_list_nat_o(null_nat,Xs) ) ).

% is_empty_set
tff(fact_8654_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_8655_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_8656_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_8657_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_8658_bit_Oconj__zero__left,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),Xa) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_8659_bit_Oconj__zero__right,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_8660_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_8661_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_8662_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_8663_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_8664_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_8665_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se8568078237143864401it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_8666_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_8667_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se8568078237143864401it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_8668_drop__bit__of__0,axiom,
    ! [Na: nat] : bit_se8568078237143864401it_int(Na,zero_zero_int) = zero_zero_int ).

% drop_bit_of_0
tff(fact_8669_drop__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% drop_bit_of_0
tff(fact_8670_drop__bit__of__bool,axiom,
    ! [Na: nat,B2: $o] :
      bit_se8568078237143864401it_int(Na,aa_o_int(zero_n2684676970156552555ol_int,(B2))) = aa_o_int(zero_n2684676970156552555ol_int,
        ( ( Na = zero_zero_nat )
        & (B2) )) ).

% drop_bit_of_bool
tff(fact_8671_drop__bit__of__bool,axiom,
    ! [Na: nat,B2: $o] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,(B2))) = aa_o_nat(zero_n2687167440665602831ol_nat,
        ( ( Na = zero_zero_nat )
        & (B2) )) ).

% drop_bit_of_bool
tff(fact_8672_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_8673_drop__bit__of__1,axiom,
    ! [Na: nat] : bit_se8568078237143864401it_int(Na,one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_8674_drop__bit__of__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_8675_and__numerals_I1_J,axiom,
    ! [Ya: num] : bit_se3949692690581998587nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,Ya))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_8676_and__numerals_I1_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ya))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_8677_and__numerals_I1_J,axiom,
    ! [Ya: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ya))) = zero_zero_int ).

% and_numerals(1)
tff(fact_8678_and__numerals_I5_J,axiom,
    ! [Xa: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit0,Xa)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_8679_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_8680_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_8681_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_8682_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_8683_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(4)
tff(fact_8684_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_8685_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(3)
tff(fact_8686_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_8687_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_8688_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_8689_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_8690_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_8691_or__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_8692_or__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B2 = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_8693_bit_Odisj__zero__right,axiom,
    ! [Xa: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),zero_zero_int) = Xa ).

% bit.disj_zero_right
tff(fact_8694_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,Xa: code_integer,Ya: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,Xa) = zero_z3403309356797280102nteger )
     => ( ( bit_se1080825931792720795nteger(A,Xa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( bit_se3949692690581998587nteger(A,Ya) = zero_z3403309356797280102nteger )
         => ( ( bit_se1080825931792720795nteger(A,Ya) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( Xa = Ya ) ) ) ) ) ).

% bit.complement_unique
tff(fact_8695_bit_Ocomplement__unique,axiom,
    ! [A: int,Xa: int,Ya: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Xa) = zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Xa) = uminus_uminus_int(one_one_int) )
       => ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Ya) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Ya) = uminus_uminus_int(one_one_int) )
           => ( Xa = Ya ) ) ) ) ) ).

% bit.complement_unique
tff(fact_8696_XOR__lower,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Ya)) ) ) ).

% XOR_lower
tff(fact_8697_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),A) = zero_zero_nat ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_8698_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = A )
    <=> ( bit_se8568078237143864401it_int(Na,A) = zero_zero_int ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_8699_XOR__upper,axiom,
    ! [Xa: int,Na: nat,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Ya)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% XOR_upper
tff(fact_8700_or__nat__unfold,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),Na) = $ite(
        M2 = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M2,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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_8701_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_8702_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_8703_bit__rec,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_8704_bit__rec,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_8705_bit__rec,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_8706_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B2: code_integer,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_8707_bit__sum__mult__2__cases,axiom,
    ! [A: int,B2: int,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_8708_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B2: nat,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_8709_max_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),A) = A ).

% max.idem
tff(fact_8710_max_Oleft__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) ).

% max.left_idem
tff(fact_8711_max_Oright__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) ).

% max.right_idem
tff(fact_8712_max_Oabsorb1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = A ) ) ).

% max.absorb1
tff(fact_8713_max_Oabsorb1,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( aa_num_num(ord_max_num(A),B2) = A ) ) ).

% max.absorb1
tff(fact_8714_max_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ).

% max.absorb1
tff(fact_8715_max_Oabsorb1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ).

% max.absorb1
tff(fact_8716_max_Oabsorb2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_8717_max_Oabsorb2,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_num(ord_max_num(A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_8718_max_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_8719_max_Oabsorb2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) ) ).

% max.absorb2
tff(fact_8720_max_Obounded__iff,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B2),C2)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_8721_max_Obounded__iff,axiom,
    ! [B2: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.bounded_iff
tff(fact_8722_max_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A: 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)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_8723_max_Obounded__iff,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.bounded_iff
tff(fact_8724_bit__0__eq,axiom,
    bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_8725_bit__0__eq,axiom,
    aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_8726_max_Oabsorb3,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B2) = A ) ) ).

% max.absorb3
tff(fact_8727_max_Oabsorb3,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = A ) ) ).

% max.absorb3
tff(fact_8728_max_Oabsorb3,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( aa_num_num(ord_max_num(A),B2) = A ) ) ).

% max.absorb3
tff(fact_8729_max_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ).

% max.absorb3
tff(fact_8730_max_Oabsorb3,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ).

% max.absorb3
tff(fact_8731_max_Oabsorb4,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_8732_max_Oabsorb4,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_8733_max_Oabsorb4,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_num_num(ord_max_num(A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_8734_max_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_8735_max_Oabsorb4,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) ) ).

% max.absorb4
tff(fact_8736_max__less__iff__conj,axiom,
    ! [Xa: 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,Xa),Ya)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_8737_max__less__iff__conj,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_8738_max__less__iff__conj,axiom,
    ! [Xa: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(Xa),Ya)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_8739_max__less__iff__conj,axiom,
    ! [Xa: 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,Xa),Ya)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_8740_max__less__iff__conj,axiom,
    ! [Xa: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_8741_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_8742_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_8743_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_8744_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_8745_max__bot,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),Xa) = Xa ).

% max_bot
tff(fact_8746_max__bot,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,bot_bot_set_int),Xa) = Xa ).

% max_bot
tff(fact_8747_max__bot,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(ord_max_set_o,bot_bot_set_o),Xa) = Xa ).

% max_bot
tff(fact_8748_max__bot,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_max_filter_nat,bot_bot_filter_nat),Xa) = Xa ).

% max_bot
tff(fact_8749_max__bot,axiom,
    ! [Xa: set_set_nat] : ord_max_set_set_nat(bot_bot_set_set_nat,Xa) = Xa ).

% max_bot
tff(fact_8750_max__bot,axiom,
    ! [Xa: set_real] : ord_max_set_real(bot_bot_set_real,Xa) = Xa ).

% max_bot
tff(fact_8751_max__bot,axiom,
    ! [Xa: set_Extended_enat] : ord_ma4205026669011143323d_enat(bot_bo7653980558646680370d_enat,Xa) = Xa ).

% max_bot
tff(fact_8752_max__bot,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xa) = Xa ).

% max_bot
tff(fact_8753_max__bot2,axiom,
    ! [Xa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,Xa),bot_bot_set_nat) = Xa ).

% max_bot2
tff(fact_8754_max__bot2,axiom,
    ! [Xa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,Xa),bot_bot_set_int) = Xa ).

% max_bot2
tff(fact_8755_max__bot2,axiom,
    ! [Xa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(ord_max_set_o,Xa),bot_bot_set_o) = Xa ).

% max_bot2
tff(fact_8756_max__bot2,axiom,
    ! [Xa: filter_nat] : aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_max_filter_nat,Xa),bot_bot_filter_nat) = Xa ).

% max_bot2
tff(fact_8757_max__bot2,axiom,
    ! [Xa: set_set_nat] : ord_max_set_set_nat(Xa,bot_bot_set_set_nat) = Xa ).

% max_bot2
tff(fact_8758_max__bot2,axiom,
    ! [Xa: set_real] : ord_max_set_real(Xa,bot_bot_set_real) = Xa ).

% max_bot2
tff(fact_8759_max__bot2,axiom,
    ! [Xa: set_Extended_enat] : ord_ma4205026669011143323d_enat(Xa,bot_bo7653980558646680370d_enat) = Xa ).

% max_bot2
tff(fact_8760_max__bot2,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),bot_bot_nat) = Xa ).

% max_bot2
tff(fact_8761_max__Suc__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),Na)) ).

% max_Suc_Suc
tff(fact_8762_max__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ).

% max_0R
tff(fact_8763_max__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ).

% max_0L
tff(fact_8764_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_8765_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_8766_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_8767_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_8768_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_8769_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = $ite(aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_8770_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),numera6620942414471956472nteger(U)) ).

% max_number_of(1)
tff(fact_8771_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_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_8772_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_8773_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_8774_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(3)
tff(fact_8775_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,Xa)) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(3)
tff(fact_8776_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(3)
tff(fact_8777_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xa)) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(3)
tff(fact_8778_max__0__1_I3_J,axiom,
    ! [Xa: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(Xa)) = numera1916890842035813515d_enat(Xa) ).

% max_0_1(3)
tff(fact_8779_max__0__1_I3_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Xa)) = numera6620942414471956472nteger(Xa) ).

% max_0_1(3)
tff(fact_8780_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,Xa)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(4)
tff(fact_8781_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xa)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(4)
tff(fact_8782_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xa)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(4)
tff(fact_8783_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xa)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(4)
tff(fact_8784_max__0__1_I4_J,axiom,
    ! [Xa: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xa),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xa) ).

% max_0_1(4)
tff(fact_8785_max__0__1_I4_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(numera6620942414471956472nteger(Xa),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(Xa) ).

% max_0_1(4)
tff(fact_8786_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_8787_max__0__1_I1_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_8788_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_8789_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_8790_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_8791_max__0__1_I2_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_8792_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_8793_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_8794_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(5)
tff(fact_8795_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,Xa)) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(5)
tff(fact_8796_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(5)
tff(fact_8797_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(5)
tff(fact_8798_max__0__1_I5_J,axiom,
    ! [Xa: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xa)) = numera1916890842035813515d_enat(Xa) ).

% max_0_1(5)
tff(fact_8799_max__0__1_I5_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(one_one_Code_integer,numera6620942414471956472nteger(Xa)) = numera6620942414471956472nteger(Xa) ).

% max_0_1(5)
tff(fact_8800_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,Xa)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(6)
tff(fact_8801_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xa)),one_one_real) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(6)
tff(fact_8802_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(6)
tff(fact_8803_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xa)),one_one_int) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(6)
tff(fact_8804_max__0__1_I6_J,axiom,
    ! [Xa: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xa),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xa) ).

% max_0_1(6)
tff(fact_8805_max__0__1_I6_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(numera6620942414471956472nteger(Xa),one_one_Code_integer) = numera6620942414471956472nteger(Xa) ).

% max_0_1(6)
tff(fact_8806_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_8807_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_8808_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_8809_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_8810_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_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_8811_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,U))),uminus_uminus_int(aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,U))),uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)),uminus_uminus_int(aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_8812_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_8813_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_8814_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_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_8815_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_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,uminus_uminus_int(aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),uminus_uminus_int(aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_8816_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_8817_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_8818_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_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_8819_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),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)),uminus_uminus_int(aa_num_int(numeral_numeral_int,V))),uminus_uminus_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_8820_bit__0,axiom,
    ! [A: code_integer] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),zero_zero_nat)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_8821_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),zero_zero_nat)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_8822_bit__0,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),zero_zero_nat)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_8823_bit__mod__2__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_8824_bit__mod__2__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_8825_bit__mod__2__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_8826_max__def,axiom,
    ! [A: set_int,B2: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,A),B2) = $ite(aa_set_int_o(ord_less_eq_set_int(A),B2),B2,A) ).

% max_def
tff(fact_8827_max__def,axiom,
    ! [A: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2),B2,A) ).

% max_def
tff(fact_8828_max__def,axiom,
    ! [A: num,B2: num] :
      aa_num_num(ord_max_num(A),B2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2),B2,A) ).

% max_def
tff(fact_8829_max__def,axiom,
    ! [A: nat,B2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2),B2,A) ).

% max_def
tff(fact_8830_max__def,axiom,
    ! [A: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2),B2,A) ).

% max_def
tff(fact_8831_max__absorb1,axiom,
    ! [Ya: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ya),Xa)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,Xa),Ya) = Xa ) ) ).

% max_absorb1
tff(fact_8832_max__absorb1,axiom,
    ! [Ya: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xa)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya) = Xa ) ) ).

% max_absorb1
tff(fact_8833_max__absorb1,axiom,
    ! [Ya: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xa)
     => ( aa_num_num(ord_max_num(Xa),Ya) = Xa ) ) ).

% max_absorb1
tff(fact_8834_max__absorb1,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ya) = Xa ) ) ).

% max_absorb1
tff(fact_8835_max__absorb1,axiom,
    ! [Ya: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xa)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya) = Xa ) ) ).

% max_absorb1
tff(fact_8836_max__absorb2,axiom,
    ! [Xa: set_int,Ya: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Ya)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,Xa),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_8837_max__absorb2,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_8838_max__absorb2,axiom,
    ! [Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Ya)
     => ( aa_num_num(ord_max_num(Xa),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_8839_max__absorb2,axiom,
    ! [Xa: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ya)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_8840_max__absorb2,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Ya)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_8841_max_OcoboundedI2,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ) ).

% max.coboundedI2
tff(fact_8842_max_OcoboundedI2,axiom,
    ! [C2: num,B2: num,A: 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(A),B2)) ) ).

% max.coboundedI2
tff(fact_8843_max_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A: 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,A),B2)) ) ).

% max.coboundedI2
tff(fact_8844_max_OcoboundedI2,axiom,
    ! [C2: int,B2: int,A: 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(aa_int_fun_int_int(ord_max_int,A),B2)) ) ).

% max.coboundedI2
tff(fact_8845_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ) ).

% max.coboundedI1
tff(fact_8846_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B2)) ) ).

% max.coboundedI1
tff(fact_8847_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)) ) ).

% max.coboundedI1
tff(fact_8848_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2)) ) ).

% max.coboundedI1
tff(fact_8849_max_Oabsorb__iff2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_8850_max_Oabsorb__iff2,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
    <=> ( aa_num_num(ord_max_num(A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_8851_max_Oabsorb__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_8852_max_Oabsorb__iff2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_8853_max_Oabsorb__iff1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_8854_max_Oabsorb__iff1,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
    <=> ( aa_num_num(ord_max_num(A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_8855_max_Oabsorb__iff1,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_8856_max_Oabsorb__iff1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) = A ) ) ).

% max.absorb_iff1
tff(fact_8857_le__max__iff__disj,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_8858_le__max__iff__disj,axiom,
    ! [Z: num,Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(ord_max_num(Xa),Ya))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_8859_le__max__iff__disj,axiom,
    ! [Z: nat,Xa: 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,Xa),Ya))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_8860_le__max__iff__disj,axiom,
    ! [Z: int,Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_8861_max_Ocobounded2,axiom,
    ! [B2: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ).

% max.cobounded2
tff(fact_8862_max_Ocobounded2,axiom,
    ! [B2: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),aa_num_num(ord_max_num(A),B2)) ).

% max.cobounded2
tff(fact_8863_max_Ocobounded2,axiom,
    ! [B2: nat,A: 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,A),B2)) ).

% max.cobounded2
tff(fact_8864_max_Ocobounded2,axiom,
    ! [B2: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2)) ).

% max.cobounded2
tff(fact_8865_max_Ocobounded1,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ).

% max.cobounded1
tff(fact_8866_max_Ocobounded1,axiom,
    ! [A: num,B2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(ord_max_num(A),B2)) ).

% max.cobounded1
tff(fact_8867_max_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)) ).

% max.cobounded1
tff(fact_8868_max_Ocobounded1,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2)) ).

% max.cobounded1
tff(fact_8869_max_Oorder__iff,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) ) ) ).

% max.order_iff
tff(fact_8870_max_Oorder__iff,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
    <=> ( A = aa_num_num(ord_max_num(A),B2) ) ) ).

% max.order_iff
tff(fact_8871_max_Oorder__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) ) ) ).

% max.order_iff
tff(fact_8872_max_Oorder__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) ) ) ).

% max.order_iff
tff(fact_8873_max_OboundedI,axiom,
    ! [B2: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B2),C2)),A) ) ) ).

% max.boundedI
tff(fact_8874_max_OboundedI,axiom,
    ! [B2: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A) ) ) ).

% max.boundedI
tff(fact_8875_max_OboundedI,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => 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)),A) ) ) ).

% max.boundedI
tff(fact_8876_max_OboundedI,axiom,
    ! [B2: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A) ) ) ).

% max.boundedI
tff(fact_8877_max_OboundedE,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B2),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.boundedE
tff(fact_8878_max_OboundedE,axiom,
    ! [B2: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.boundedE
tff(fact_8879_max_OboundedE,axiom,
    ! [B2: nat,C2: nat,A: 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)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.boundedE
tff(fact_8880_max_OboundedE,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.boundedE
tff(fact_8881_max_OorderI,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% max.orderI
tff(fact_8882_max_OorderI,axiom,
    ! [A: num,B2: num] :
      ( ( A = aa_num_num(ord_max_num(A),B2) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ).

% max.orderI
tff(fact_8883_max_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ).

% max.orderI
tff(fact_8884_max_OorderI,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% max.orderI
tff(fact_8885_max_OorderE,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) ) ) ).

% max.orderE
tff(fact_8886_max_OorderE,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( A = aa_num_num(ord_max_num(A),B2) ) ) ).

% max.orderE
tff(fact_8887_max_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) ) ) ).

% max.orderE
tff(fact_8888_max_OorderE,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) ) ) ).

% max.orderE
tff(fact_8889_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,C2),D)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ) ) ).

% max.mono
tff(fact_8890_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(C2),D)),aa_num_num(ord_max_num(A),B2)) ) ) ).

% max.mono
tff(fact_8891_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),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),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)) ) ) ).

% max.mono
tff(fact_8892_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C2),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2)) ) ) ).

% max.mono
tff(fact_8893_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B2: real,A: 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,A),B2)) ) ).

% max.strict_coboundedI2
tff(fact_8894_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B2: rat,A: 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(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ) ).

% max.strict_coboundedI2
tff(fact_8895_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B2: num,A: 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(A),B2)) ) ).

% max.strict_coboundedI2
tff(fact_8896_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A: 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,A),B2)) ) ).

% max.strict_coboundedI2
tff(fact_8897_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B2: int,A: 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(aa_int_fun_int_int(ord_max_int,A),B2)) ) ).

% max.strict_coboundedI2
tff(fact_8898_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B2)) ) ).

% max.strict_coboundedI1
tff(fact_8899_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2)) ) ).

% max.strict_coboundedI1
tff(fact_8900_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(ord_max_num(A),B2)) ) ).

% max.strict_coboundedI1
tff(fact_8901_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)) ) ).

% max.strict_coboundedI1
tff(fact_8902_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2)) ) ).

% max.strict_coboundedI1
tff(fact_8903_max_Ostrict__order__iff,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_8904_max_Ostrict__order__iff,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_8905_max_Ostrict__order__iff,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
    <=> ( ( A = aa_num_num(ord_max_num(A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_8906_max_Ostrict__order__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_8907_max_Ostrict__order__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B2) )
        & ( A != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_8908_max_Ostrict__boundedE,axiom,
    ! [B2: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B2),C2)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_8909_max_Ostrict__boundedE,axiom,
    ! [B2: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B2),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_8910_max_Ostrict__boundedE,axiom,
    ! [B2: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(B2),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_8911_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A: 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)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_8912_max_Ostrict__boundedE,axiom,
    ! [B2: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_8913_less__max__iff__disj,axiom,
    ! [Z: real,Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Ya))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_8914_less__max__iff__disj,axiom,
    ! [Z: rat,Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_8915_less__max__iff__disj,axiom,
    ! [Z: num,Xa: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(ord_max_num(Xa),Ya))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xa)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_8916_less__max__iff__disj,axiom,
    ! [Z: nat,Xa: 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,Xa),Ya))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xa)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_8917_less__max__iff__disj,axiom,
    ! [Z: int,Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xa)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_8918_of__nat__max,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(semiri1316708129612266289at_nat,Ya)) ).

% of_nat_max
tff(fact_8919_of__nat__max,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ya)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Ya)) ).

% of_nat_max
tff(fact_8920_of__nat__max,axiom,
    ! [Xa: nat,Ya: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(semiri5074537144036343181t_real,Ya)) ).

% of_nat_max
tff(fact_8921_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_8922_max_Oassoc,axiom,
    ! [A: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)) ).

% max.assoc
tff(fact_8923_max_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),A) ).

% max.commute
tff(fact_8924_max_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)) ).

% max.left_commute
tff(fact_8925_sup__max,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_max
tff(fact_8926_max__diff__distrib__left,axiom,
    ! [Xa: real,Ya: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Ya)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Z)) ).

% max_diff_distrib_left
tff(fact_8927_max__diff__distrib__left,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Ya),Z)) ).

% max_diff_distrib_left
tff(fact_8928_max__diff__distrib__left,axiom,
    ! [Xa: int,Ya: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ya),Z)) ).

% max_diff_distrib_left
tff(fact_8929_max__add__distrib__right,axiom,
    ! [Xa: real,Ya: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa),Ya)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Z)) ).

% max_add_distrib_right
tff(fact_8930_max__add__distrib__right,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Ya),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Ya)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Z)) ).

% max_add_distrib_right
tff(fact_8931_max__add__distrib__right,axiom,
    ! [Xa: int,Ya: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(aa_int_fun_int_int(ord_max_int,Ya),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Ya)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)) ).

% max_add_distrib_right
tff(fact_8932_max__add__distrib__right,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),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,Xa),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)) ).

% max_add_distrib_right
tff(fact_8933_max__add__distrib__left,axiom,
    ! [Xa: real,Ya: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Ya)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ya),Z)) ).

% max_add_distrib_left
tff(fact_8934_max__add__distrib__left,axiom,
    ! [Xa: rat,Ya: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Xa),Ya)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Ya),Z)) ).

% max_add_distrib_left
tff(fact_8935_max__add__distrib__left,axiom,
    ! [Xa: int,Ya: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xa),Ya)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ya),Z)) ).

% max_add_distrib_left
tff(fact_8936_max__add__distrib__left,axiom,
    ! [Xa: 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,Xa),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ya),Z)) ).

% max_add_distrib_left
tff(fact_8937_nat__mult__max__left,axiom,
    ! [M2: nat,Na: nat,Q5: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),Na)),Q5) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q5)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)) ).

% nat_mult_max_left
tff(fact_8938_nat__mult__max__right,axiom,
    ! [M2: nat,Na: nat,Q5: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q5)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q5)) ).

% nat_mult_max_right
tff(fact_8939_nat__add__max__left,axiom,
    ! [M2: nat,Na: nat,Q5: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),Na)),Q5) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q5)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Q5)) ).

% nat_add_max_left
tff(fact_8940_nat__add__max__right,axiom,
    ! [M2: nat,Na: nat,Q5: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q5)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q5)) ).

% nat_add_max_right
tff(fact_8941_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Na)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ) ).

% bit_nat_iff
tff(fact_8942_bit__1__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_8943_bit__1__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_8944_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_8945_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_8946_bit__take__bit__iff,axiom,
    ! [M2: nat,A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ) ).

% bit_take_bit_iff
tff(fact_8947_bit__take__bit__iff,axiom,
    ! [M2: nat,A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(M2),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
        & aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ) ).

% bit_take_bit_iff
tff(fact_8948_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) ) ).

% or_greater_eq
tff(fact_8949_OR__lower,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Ya)) ) ) ).

% OR_lower
tff(fact_8950_bit__of__bool__iff,axiom,
    ! [B2: $o,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_o_int(zero_n2684676970156552555ol_int,(B2))),Na)
    <=> ( (B2)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_8951_bit__of__bool__iff,axiom,
    ! [B2: $o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),Na)
    <=> ( (B2)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_8952_AND__upper2_H,axiom,
    ! [Ya: int,Z: int,Xa: 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(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Ya)),Z) ) ) ).

% AND_upper2'
tff(fact_8953_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(aa_int_fun_int_int(bit_se725231765392027082nd_int,Ya),Yaa)),Z) ) ) ).

% AND_upper1'
tff(fact_8954_AND__upper2,axiom,
    ! [Ya: int,Xa: 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(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Ya)),Ya) ) ).

% AND_upper2
tff(fact_8955_AND__upper1,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Ya)),Xa) ) ).

% AND_upper1
tff(fact_8956_AND__lower,axiom,
    ! [Xa: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Ya)) ) ).

% AND_lower
tff(fact_8957_nat__minus__add__max,axiom,
    ! [Na: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),M2) ).

% nat_minus_add_max
tff(fact_8958_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8959_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_8960_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_8961_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Na)) ).

% not_bit_Suc_0_numeral
tff(fact_8962_AND__upper2_H_H,axiom,
    ! [Ya: int,Z: int,Xa: 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(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Ya)),Z) ) ) ).

% AND_upper2''
tff(fact_8963_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(aa_int_fun_int_int(bit_se725231765392027082nd_int,Ya),Yaa)),Z) ) ) ).

% AND_upper1''
tff(fact_8964_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K) ) ).

% and_less_eq
tff(fact_8965_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M2: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M2),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8966_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_8967_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_8968_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: code_integer] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na) = zero_z3403309356797280102nteger )
     => ~ aa_nat_o(bit_se9216721137139052372nteger(A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_8969_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_8970_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_8971_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M3)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M3)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K),N2) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N2) ) ) ) ).

% int_bit_bound
tff(fact_8972_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_8973_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_8974_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( bit_se3949692690581998587nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Na)) = zero_z3403309356797280102nteger )
    <=> ~ aa_nat_o(bit_se9216721137139052372nteger(A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_8975_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_8976_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_8977_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_8978_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_8979_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_8980_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_8981_even__bit__succ__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A)),Na)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_8982_even__bit__succ__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Na)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_8983_even__bit__succ__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_8984_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,Na: nat] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_8985_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Na: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_8986_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_8987_OR__upper,axiom,
    ! [Xa: int,Na: nat,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Ya)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% OR_upper
tff(fact_8988_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_8989_cot__less__zero,axiom,
    ! [Xa: 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)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xa)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_8990_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),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_8991_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),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_8992_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) ) ) ).

% cot_gt_zero
tff(fact_8993_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_8994_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_8995_cot__zero,axiom,
    aa_real_real(cot_real,zero_zero_real) = zero_zero_real ).

% cot_zero
tff(fact_8996_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8997_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8998_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_8999_cos__one__sin__zero,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xa) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_9000_cos__one__sin__zero,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = one_one_real )
     => ( aa_real_real(sin_real,Xa) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_9001_sin__zero__norm__cos__one,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xa)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_9002_sin__zero__norm__cos__one,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(sin_complex,Xa) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xa)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_9003_sincos__principal__value,axiom,
    ! [Xa: 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,Xa) )
      & ( aa_real_real(cos_real,Y) = aa_real_real(cos_real,Xa) ) ) ).

% sincos_principal_value
tff(fact_9004_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_9005_sin__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),one_one_real) ).

% sin_le_one
tff(fact_9006_cos__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),one_one_real) ).

% cos_le_one
tff(fact_9007_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9008_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xa))),aa_real_real(abs_abs_real,Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_9009_sin__cos__le1,axiom,
    ! [Xa: 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,Xa)),aa_real_real(sin_real,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Ya))))),one_one_real) ).

% sin_cos_le1
tff(fact_9010_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_9011_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(sin_real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_9012_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_9013_sin__ge__minus__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xa)) ).

% sin_ge_minus_one
tff(fact_9014_cos__inj__pi,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),pi)
           => ( ( aa_real_real(cos_real,Xa) = aa_real_real(cos_real,Ya) )
             => ( Xa = Ya ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_9015_cos__mono__le__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa)),aa_real_real(cos_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_9016_cos__monotone__0__pi__le,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Ya)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_9017_cos__ge__minus__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xa)) ).

% cos_ge_minus_one
tff(fact_9018_abs__sin__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xa))),one_one_real) ).

% abs_sin_le_one
tff(fact_9019_abs__cos__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xa))),one_one_real) ).

% abs_cos_le_one
tff(fact_9020_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(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,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_9021_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xa)) ) ) ) ).

% arcsin_minus
tff(fact_9022_arcsin__le__arcsin,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),aa_real_real(arcsin,Ya)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9023_arcsin__eq__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( ( aa_real_real(arcsin,Xa) = aa_real_real(arcsin,Ya) )
        <=> ( Xa = Ya ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9024_arcsin__le__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ) ).

% arcsin_le_mono
tff(fact_9025_cos__mono__less__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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,Xa)),aa_real_real(cos_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_9026_cos__monotone__0__pi,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Ya)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_9027_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
         => ( Xa = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_9028_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),pi)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_9029_cos__monotone__minus__pi__0_H,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Ya)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_9030_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_9031_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_9032_arcsin__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(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,Xa)),Ya)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sin_real,Ya)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9033_le__arcsin__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(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,Xa))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Ya)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9034_sincos__total__pi,axiom,
    ! [Ya: real,Xa: 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,Xa),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 )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),pi)
            & ( Xa = aa_real_real(cos_real,T4) )
            & ( Ya = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_9035_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(sqrt,aa_real_real(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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9036_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_9037_arcsin__less__arcsin,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)),aa_real_real(arcsin,Ya)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9038_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_9039_arcsin__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ) ).

% arcsin_less_mono
tff(fact_9040_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_9041_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y2: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y2) = zero_zero_real ) )
         => ( Y2 = X2 ) ) ) ).

% cos_is_zero
tff(fact_9042_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_9043_cos__monotone__minus__pi__0,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Ya)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_9044_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)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),pi)
            & ( aa_real_real(cos_real,X2) = Ya )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
                  & ( aa_real_real(cos_real,Y2) = Ya ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_9045_sincos__total__pi__half,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xa = aa_real_real(cos_real,T4) )
              & ( Ya = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9046_sincos__total__2pi__le,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xa = aa_real_real(cos_real,T4) )
          & ( Ya = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9047_sincos__total__2pi,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xa = aa_real_real(cos_real,T4) )
               => ( Ya != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_9048_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_9049_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_9050_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_9051_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_9052_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) ) ) ).

% cos_gt_zero
tff(fact_9053_sin__monotone__2pi__le,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_9054_sin__mono__le__eq,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),aa_real_real(sin_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_9055_sin__inj__pi,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa) = aa_real_real(sin_real,Ya) )
             => ( Xa = Ya ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_9056_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_9057_sin__less__zero,axiom,
    ! [Xa: 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)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_9058_sin__monotone__2pi,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_9059_sin__mono__less__eq,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),aa_real_real(sin_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_9060_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)
       => ? [X2: 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))))),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),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,X2) = Ya )
            & ! [Y2: 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))))),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),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,Y2) = Ya ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_9061_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_9062_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(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,Xa)) ) ) ).

% cos_ge_zero
tff(fact_9063_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_9064_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_9065_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_9066_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_9067_tan__double,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)) = 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))),aa_real_real(tan_real,Xa))),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(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_9068_tan__double,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xa)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xa))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_9069_sin__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_9070_cos__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xa) = aa_real_real(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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9071_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(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,T4),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9072_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_9073_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_9074_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_9075_arccos__le__arccos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_9076_arccos__le__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xa)),aa_real_real(arccos,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_9077_arccos__eq__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real) )
     => ( ( aa_real_real(arccos,Xa) = aa_real_real(arccos,Ya) )
      <=> ( Xa = Ya ) ) ) ).

% arccos_eq_iff
tff(fact_9078_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_9079_arccos__less__arccos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_9080_arccos__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xa)),aa_real_real(arccos,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_9081_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_9082_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_9083_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_9084_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_9085_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_9086_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_9087_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9088_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xa)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = aa_real_real(uminus_uminus_real,Xa) ) ) ) ).

% arccos_cos2
tff(fact_9089_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_minus
tff(fact_9090_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) ) ) ).

% tan_gt_zero
tff(fact_9091_lemma__tan__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(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,X2)) ) ) ).

% lemma_tan_total
tff(fact_9092_lemma__tan__total1,axiom,
    ! [Ya: real] :
    ? [X2: 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))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),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,X2) = Ya ) ) ).

% lemma_tan_total1
tff(fact_9093_tan__mono__lt__eq,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)),aa_real_real(tan_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_9094_tan__monotone_H,axiom,
    ! [Ya: real,Xa: 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))))),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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),Xa)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Ya)),aa_real_real(tan_real,Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_9095_tan__monotone,axiom,
    ! [Ya: real,Xa: 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),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)) ) ) ) ).

% tan_monotone
tff(fact_9096_tan__total,axiom,
    ! [Ya: real] :
    ? [X2: 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))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),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,X2) = Ya )
      & ! [Y2: 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))))),Y2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),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,Y2) = Ya ) )
         => ( Y2 = X2 ) ) ) ).

% tan_total
tff(fact_9097_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_9098_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xa)) ) ) ).

% arccos_minus_abs
tff(fact_9099_add__tan__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Ya) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Ya))) ) ) ) ).

% add_tan_eq
tff(fact_9100_add__tan__eq,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Ya) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xa)),aa_complex_complex(tan_complex,Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xa),Ya))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xa)),aa_complex_complex(cos_complex,Ya))) ) ) ) ).

% add_tan_eq
tff(fact_9101_tan__total__pos,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X2) = Ya ) ) ) ).

% tan_total_pos
tff(fact_9102_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_9103_tan__less__zero,axiom,
    ! [Xa: 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)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xa)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_9104_tan__mono__le,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),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,Xa)),aa_real_real(tan_real,Ya)) ) ) ) ).

% tan_mono_le
tff(fact_9105_tan__mono__le__eq,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa)),aa_real_real(tan_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Ya) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9106_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(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,Xa))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_9107_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_9108_arctan__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(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,Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_9109_arctan__unique,axiom,
    ! [Xa: 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))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),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,Xa) = Ya )
         => ( aa_real_real(arctan,Ya) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_9110_lemma__tan__add1,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Ya) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Ya))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Ya))) ) ) ) ).

% lemma_tan_add1
tff(fact_9111_lemma__tan__add1,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Ya) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xa)),aa_complex_complex(tan_complex,Ya))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xa),Ya))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xa)),aa_complex_complex(cos_complex,Ya))) ) ) ) ).

% lemma_tan_add1
tff(fact_9112_tan__diff,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Ya) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Ya)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Ya)))) ) ) ) ) ).

% tan_diff
tff(fact_9113_tan__diff,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Ya) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xa),Ya)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xa),Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xa)),aa_complex_complex(tan_complex,Ya))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xa)),aa_complex_complex(tan_complex,Ya)))) ) ) ) ) ).

% tan_diff
tff(fact_9114_tan__add,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Ya) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Ya))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Ya)))) ) ) ) ) ).

% tan_add
tff(fact_9115_tan__add,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Ya) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xa),Ya)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xa),Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xa)),aa_complex_complex(tan_complex,Ya))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xa)),aa_complex_complex(tan_complex,Ya)))) ) ) ) ) ).

% tan_add
tff(fact_9116_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ? [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) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_9117_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_9118_tan__sec,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(cos_complex,Xa) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_9119_tan__sec,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(cos_real,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_9120_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_9121_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_9122_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(real_V4546457046886955230omplex(pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_9123_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_9124_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% push_bit_of_Suc_0
tff(fact_9125_set__removeAll,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(removeAll_VEBT_VEBT(Xa,Xs)) = minus_5127226145743854075T_VEBT(set_VEBT_VEBT2(Xs),insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT)) ).

% set_removeAll
tff(fact_9126_set__removeAll,axiom,
    ! [Xa: int,Xs: list_int] : set_int2(removeAll_int(Xa,Xs)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,set_int2(Xs)),insert_int(Xa,bot_bot_set_int)) ).

% set_removeAll
tff(fact_9127_set__removeAll,axiom,
    ! [Xa: $o,Xs: list_o] : set_o2(removeAll_o((Xa),Xs)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(minus_minus_set_o,set_o2(Xs)),insert_o((Xa),bot_bot_set_o)) ).

% set_removeAll
tff(fact_9128_set__removeAll,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] : set_set_nat2(removeAll_set_nat(Xa,Xs)) = minus_2163939370556025621et_nat(set_set_nat2(Xs),insert_set_nat(Xa,bot_bot_set_set_nat)) ).

% set_removeAll
tff(fact_9129_set__removeAll,axiom,
    ! [Xa: real,Xs: list_real] : set_real2(removeAll_real(Xa,Xs)) = minus_minus_set_real(set_real2(Xs),insert_real(Xa,bot_bot_set_real)) ).

% set_removeAll
tff(fact_9130_set__removeAll,axiom,
    ! [Xa: extended_enat,Xs: list_Extended_enat] : set_Extended_enat2(remove8473807646742367858d_enat(Xa,Xs)) = minus_925952699566721837d_enat(set_Extended_enat2(Xs),insert_Extended_enat(Xa,bot_bo7653980558646680370d_enat)) ).

% set_removeAll
tff(fact_9131_set__removeAll,axiom,
    ! [Xa: nat,Xs: list_nat] : set_nat2(removeAll_nat(Xa,Xs)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat(Xa,bot_bot_set_nat)) ).

% set_removeAll
tff(fact_9132_push__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% push_bit_of_0
tff(fact_9133_push__bit__of__0,axiom,
    ! [Na: nat] : bit_se545348938243370406it_int(Na,zero_zero_int) = zero_zero_int ).

% push_bit_of_0
tff(fact_9134_push__bit__eq__0__iff,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
tff(fact_9135_push__bit__eq__0__iff,axiom,
    ! [Na: nat,A: int] :
      ( ( bit_se545348938243370406it_int(Na,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
tff(fact_9136_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_9137_of__real__0,axiom,
    real_V4546457046886955230omplex(zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_9138_of__real__eq__0__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_9139_of__real__eq__0__iff,axiom,
    ! [Xa: real] :
      ( ( real_V4546457046886955230omplex(Xa) = zero_zero_complex )
    <=> ( Xa = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_9140_sin__of__real__pi,axiom,
    aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,pi)) = zero_zero_real ).

% sin_of_real_pi
tff(fact_9141_sin__of__real__pi,axiom,
    aa_complex_complex(sin_complex,real_V4546457046886955230omplex(pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_9142_even__push__bit__iff,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se7788150548672797655nteger(Na,A))
    <=> ( ( Na != zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_9143_even__push__bit__iff,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A))
    <=> ( ( Na != zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_9144_even__push__bit__iff,axiom,
    ! [Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_se545348938243370406it_int(Na,A))
    <=> ( ( Na != zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_9145_length__removeAll__less__eq,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(Xa,Xs))),size_s6755466524823107622T_VEBT(Xs)) ).

% length_removeAll_less_eq
tff(fact_9146_length__removeAll__less__eq,axiom,
    ! [Xa: $o,Xs: list_o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(removeAll_o((Xa),Xs))),size_size_list_o(Xs)) ).

% length_removeAll_less_eq
tff(fact_9147_length__removeAll__less__eq,axiom,
    ! [Xa: nat,Xs: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(removeAll_nat(Xa,Xs))),size_size_list_nat(Xs)) ).

% length_removeAll_less_eq
tff(fact_9148_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M2),Q5)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9149_norm__less__p1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xa)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xa))),one_one_real))) ).

% norm_less_p1
tff(fact_9150_norm__less__p1,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(real_V4546457046886955230omplex(aa_complex_real(real_V1022390504157884413omplex,Xa))),one_one_complex))) ).

% norm_less_p1
tff(fact_9151_length__removeAll__less,axiom,
    ! [Xa: extended_enat,Xs: list_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),set_Extended_enat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s3941691890525107288d_enat(remove8473807646742367858d_enat(Xa,Xs))),size_s3941691890525107288d_enat(Xs)) ) ).

% length_removeAll_less
tff(fact_9152_length__removeAll__less,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(removeAll_real(Xa,Xs))),size_size_list_real(Xs)) ) ).

% length_removeAll_less
tff(fact_9153_length__removeAll__less,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s3254054031482475050et_nat(removeAll_set_nat(Xa,Xs))),size_s3254054031482475050et_nat(Xs)) ) ).

% length_removeAll_less
tff(fact_9154_length__removeAll__less,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(removeAll_int(Xa,Xs))),size_size_list_int(Xs)) ) ).

% length_removeAll_less
tff(fact_9155_length__removeAll__less,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(Xa,Xs))),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_removeAll_less
tff(fact_9156_length__removeAll__less,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(removeAll_o((Xa),Xs))),size_size_list_o(Xs)) ) ).

% length_removeAll_less
tff(fact_9157_length__removeAll__less,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(removeAll_nat(Xa,Xs))),size_size_list_nat(Xs)) ) ).

% length_removeAll_less
tff(fact_9158_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),one_one_nat)) != zero_zero_nat ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_9159_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),bit_se545348938243370406it_int(Na,one_one_int)) != zero_zero_int ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_9160_norm__of__real__diff,axiom,
    ! [B2: real,A: 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(real_V1803761363581548252l_real,B2)),aa_real_real(real_V1803761363581548252l_real,A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))) ).

% norm_of_real_diff
tff(fact_9161_norm__of__real__diff,axiom,
    ! [B2: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(real_V4546457046886955230omplex(B2)),real_V4546457046886955230omplex(A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))) ).

% norm_of_real_diff
tff(fact_9162_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(groups3417619833198082522nteger(zero_n356916108424825756nteger,numera6620942414471956472nteger(aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_9163_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,groups9119017779487936845_o_nat(zero_n2687167440665602831ol_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_9164_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_9165_inthall,axiom,
    ! [Xs: list_Extended_enat,P: fun_Extended_enat_o,Na: nat] :
      ( ! [X2: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(X2),set_Extended_enat2(Xs))
         => aa_Extended_enat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3941691890525107288d_enat(Xs))
       => aa_Extended_enat_o(P,nth_Extended_enat(Xs,Na)) ) ) ).

% inthall
tff(fact_9166_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Na: nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Na)) ) ) ).

% inthall
tff(fact_9167_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Na: nat] :
      ( ! [X2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X2),set_set_nat2(Xs))
         => aa_set_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,nth_set_nat(Xs,Na)) ) ) ).

% inthall
tff(fact_9168_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Na: nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_int_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% inthall
tff(fact_9169_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Na: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% inthall
tff(fact_9170_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Na: nat] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),set_o2(Xs))
         => aa_o_o(P,(X2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% inthall
tff(fact_9171_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Na: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_9172_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_9173_sub__num__simps_I3_J,axiom,
    ! [L: num] : aa_num_int(neg_numeral_sub_int(one),aa_num_num(bit1,L)) = uminus_uminus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,L))) ).

% sub_num_simps(3)
tff(fact_9174_sub__num__simps_I3_J,axiom,
    ! [L: num] : aa_num_real(neg_numeral_sub_real(one),aa_num_num(bit1,L)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,L))) ).

% sub_num_simps(3)
tff(fact_9175_sub__num__simps_I3_J,axiom,
    ! [L: num] : aa_num_rat(neg_numeral_sub_rat(one),aa_num_num(bit1,L)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,L))) ).

% sub_num_simps(3)
tff(fact_9176_sub__num__simps_I3_J,axiom,
    ! [L: num] : neg_nu5755505904847501662nteger(one,aa_num_num(bit1,L)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,L))) ).

% sub_num_simps(3)
tff(fact_9177_sub__num__simps_I3_J,axiom,
    ! [L: num] : neg_nu8416839295433526191omplex(one,aa_num_num(bit1,L)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,L))) ).

% sub_num_simps(3)
tff(fact_9178_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_9179_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Na)),TreeList2,S4) ) ).

% deg_SUcn_Node
tff(fact_9180_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se545348938243370406it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9181_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se545348938243370406it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9182_sub__num__simps_I1_J,axiom,
    aa_num_real(neg_numeral_sub_real(one),one) = zero_zero_real ).

% sub_num_simps(1)
tff(fact_9183_sub__num__simps_I1_J,axiom,
    aa_num_rat(neg_numeral_sub_rat(one),one) = zero_zero_rat ).

% sub_num_simps(1)
tff(fact_9184_sub__num__simps_I1_J,axiom,
    aa_num_int(neg_numeral_sub_int(one),one) = zero_zero_int ).

% sub_num_simps(1)
tff(fact_9185_diff__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(neg_numeral_sub_rat(M2),Na) ).

% diff_numeral_simps(1)
tff(fact_9186_diff__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(neg_numeral_sub_real(M2),Na) ).

% diff_numeral_simps(1)
tff(fact_9187_diff__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(neg_numeral_sub_int(M2),Na) ).

% diff_numeral_simps(1)
tff(fact_9188_diff__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(Na)) = neg_nu5755505904847501662nteger(M2,Na) ).

% diff_numeral_simps(1)
tff(fact_9189_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(neg_numeral_sub_int(M2),Na) ).

% add_neg_numeral_simps(1)
tff(fact_9190_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(neg_numeral_sub_real(M2),Na) ).

% add_neg_numeral_simps(1)
tff(fact_9191_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(neg_numeral_sub_rat(M2),Na) ).

% add_neg_numeral_simps(1)
tff(fact_9192_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(M2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = neg_nu5755505904847501662nteger(M2,Na) ).

% add_neg_numeral_simps(1)
tff(fact_9193_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M2)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = neg_nu8416839295433526191omplex(M2,Na) ).

% add_neg_numeral_simps(1)
tff(fact_9194_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(neg_numeral_sub_int(Na),M2) ).

% add_neg_numeral_simps(2)
tff(fact_9195_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(neg_numeral_sub_real(Na),M2) ).

% add_neg_numeral_simps(2)
tff(fact_9196_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(neg_numeral_sub_rat(Na),M2) ).

% add_neg_numeral_simps(2)
tff(fact_9197_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(Na)) = neg_nu5755505904847501662nteger(Na,M2) ).

% add_neg_numeral_simps(2)
tff(fact_9198_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),numera6690914467698888265omplex(Na)) = neg_nu8416839295433526191omplex(Na,M2) ).

% add_neg_numeral_simps(2)
tff(fact_9199_diff__numeral__simps_I4_J,axiom,
    ! [M2: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(neg_numeral_sub_int(Na),M2) ).

% diff_numeral_simps(4)
tff(fact_9200_diff__numeral__simps_I4_J,axiom,
    ! [M2: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(neg_numeral_sub_real(Na),M2) ).

% diff_numeral_simps(4)
tff(fact_9201_diff__numeral__simps_I4_J,axiom,
    ! [M2: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(neg_numeral_sub_rat(Na),M2) ).

% diff_numeral_simps(4)
tff(fact_9202_diff__numeral__simps_I4_J,axiom,
    ! [M2: num,Na: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = neg_nu5755505904847501662nteger(Na,M2) ).

% diff_numeral_simps(4)
tff(fact_9203_diff__numeral__simps_I4_J,axiom,
    ! [M2: num,Na: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = neg_nu8416839295433526191omplex(Na,M2) ).

% diff_numeral_simps(4)
tff(fact_9204_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M2)),one_one_complex) = neg_nu8416839295433526191omplex(M2,one) ).

% diff_numeral_special(2)
tff(fact_9205_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,M2)),one_one_rat) = aa_num_rat(neg_numeral_sub_rat(M2),one) ).

% diff_numeral_special(2)
tff(fact_9206_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M2)),one_one_real) = aa_num_real(neg_numeral_sub_real(M2),one) ).

% diff_numeral_special(2)
tff(fact_9207_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M2)),one_one_int) = aa_num_int(neg_numeral_sub_int(M2),one) ).

% diff_numeral_special(2)
tff(fact_9208_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M2),one_one_Code_integer) = neg_nu5755505904847501662nteger(M2,one) ).

% diff_numeral_special(2)
tff(fact_9209_diff__numeral__special_I1_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(one_one_complex),numera6690914467698888265omplex(Na)) = neg_nu8416839295433526191omplex(one,Na) ).

% diff_numeral_special(1)
tff(fact_9210_diff__numeral__special_I1_J,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(neg_numeral_sub_rat(one),Na) ).

% diff_numeral_special(1)
tff(fact_9211_diff__numeral__special_I1_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(neg_numeral_sub_real(one),Na) ).

% diff_numeral_special(1)
tff(fact_9212_diff__numeral__special_I1_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(neg_numeral_sub_int(one),Na) ).

% diff_numeral_special(1)
tff(fact_9213_diff__numeral__special_I1_J,axiom,
    ! [Na: num] : minus_8373710615458151222nteger(one_one_Code_integer,numera6620942414471956472nteger(Na)) = neg_nu5755505904847501662nteger(one,Na) ).

% diff_numeral_special(1)
tff(fact_9214_sub__num__simps_I5_J,axiom,
    ! [K: num] : aa_num_real(neg_numeral_sub_real(aa_num_num(bit1,K)),one) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% sub_num_simps(5)
tff(fact_9215_sub__num__simps_I5_J,axiom,
    ! [K: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bit1,K)),one) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% sub_num_simps(5)
tff(fact_9216_sub__num__simps_I5_J,axiom,
    ! [K: num] : neg_nu5755505904847501662nteger(aa_num_num(bit1,K),one) = numera6620942414471956472nteger(aa_num_num(bit0,K)) ).

% sub_num_simps(5)
tff(fact_9217_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))) = aa_num_int(neg_numeral_sub_int(one),M2) ).

% add_neg_numeral_special(1)
tff(fact_9218_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))) = aa_num_real(neg_numeral_sub_real(one),M2) ).

% add_neg_numeral_special(1)
tff(fact_9219_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))) = aa_num_rat(neg_numeral_sub_rat(one),M2) ).

% add_neg_numeral_special(1)
tff(fact_9220_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) = neg_nu5755505904847501662nteger(one,M2) ).

% add_neg_numeral_special(1)
tff(fact_9221_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))) = neg_nu8416839295433526191omplex(one,M2) ).

% add_neg_numeral_special(1)
tff(fact_9222_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),one_one_int) = aa_num_int(neg_numeral_sub_int(one),M2) ).

% add_neg_numeral_special(2)
tff(fact_9223_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real) = aa_num_real(neg_numeral_sub_real(one),M2) ).

% add_neg_numeral_special(2)
tff(fact_9224_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),one_one_rat) = aa_num_rat(neg_numeral_sub_rat(one),M2) ).

% add_neg_numeral_special(2)
tff(fact_9225_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),one_one_Code_integer) = neg_nu5755505904847501662nteger(one,M2) ).

% add_neg_numeral_special(2)
tff(fact_9226_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),one_one_complex) = neg_nu8416839295433526191omplex(one,M2) ).

% add_neg_numeral_special(2)
tff(fact_9227_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(one_one_int)) = aa_num_int(neg_numeral_sub_int(M2),one) ).

% add_neg_numeral_special(3)
tff(fact_9228_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(neg_numeral_sub_real(M2),one) ).

% add_neg_numeral_special(3)
tff(fact_9229_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(neg_numeral_sub_rat(M2),one) ).

% add_neg_numeral_special(3)
tff(fact_9230_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(M2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(M2,one) ).

% add_neg_numeral_special(3)
tff(fact_9231_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M2)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = neg_nu8416839295433526191omplex(M2,one) ).

% add_neg_numeral_special(3)
tff(fact_9232_add__neg__numeral__special_I4_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(neg_numeral_sub_int(Na),one) ).

% add_neg_numeral_special(4)
tff(fact_9233_add__neg__numeral__special_I4_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(neg_numeral_sub_real(Na),one) ).

% add_neg_numeral_special(4)
tff(fact_9234_add__neg__numeral__special_I4_J,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(neg_numeral_sub_rat(Na),one) ).

% add_neg_numeral_special(4)
tff(fact_9235_add__neg__numeral__special_I4_J,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(Na)) = neg_nu5755505904847501662nteger(Na,one) ).

% add_neg_numeral_special(4)
tff(fact_9236_add__neg__numeral__special_I4_J,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Na)) = neg_nu8416839295433526191omplex(Na,one) ).

% add_neg_numeral_special(4)
tff(fact_9237_minus__sub__one__diff__one,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(aa_num_int(neg_numeral_sub_int(M2),one))),one_one_int) = uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)) ).

% minus_sub_one_diff_one
tff(fact_9238_minus__sub__one__diff__one,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(neg_numeral_sub_real(M2),one))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)) ).

% minus_sub_one_diff_one
tff(fact_9239_minus__sub__one__diff__one,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(neg_numeral_sub_rat(M2),one))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)) ).

% minus_sub_one_diff_one
tff(fact_9240_minus__sub__one__diff__one,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5755505904847501662nteger(M2,one)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)) ).

% minus_sub_one_diff_one
tff(fact_9241_minus__sub__one__diff__one,axiom,
    ! [M2: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,neg_nu8416839295433526191omplex(M2,one))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2)) ).

% minus_sub_one_diff_one
tff(fact_9242_diff__numeral__special_I7_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(one_one_int)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = aa_num_int(neg_numeral_sub_int(Na),one) ).

% diff_numeral_special(7)
tff(fact_9243_diff__numeral__special_I7_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(neg_numeral_sub_real(Na),one) ).

% diff_numeral_special(7)
tff(fact_9244_diff__numeral__special_I7_J,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(neg_numeral_sub_rat(Na),one) ).

% diff_numeral_special(7)
tff(fact_9245_diff__numeral__special_I7_J,axiom,
    ! [Na: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = neg_nu5755505904847501662nteger(Na,one) ).

% diff_numeral_special(7)
tff(fact_9246_diff__numeral__special_I7_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = neg_nu8416839295433526191omplex(Na,one) ).

% diff_numeral_special(7)
tff(fact_9247_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(one_one_int)) = aa_num_int(neg_numeral_sub_int(one),M2) ).

% diff_numeral_special(8)
tff(fact_9248_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(neg_numeral_sub_real(one),M2) ).

% diff_numeral_special(8)
tff(fact_9249_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(neg_numeral_sub_rat(one),M2) ).

% diff_numeral_special(8)
tff(fact_9250_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(one,M2) ).

% diff_numeral_special(8)
tff(fact_9251_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = neg_nu8416839295433526191omplex(one,M2) ).

% diff_numeral_special(8)
tff(fact_9252_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_9253_VEBT_Oexhaust,axiom,
    ! [Ya: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Ya != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Ya != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_9254_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_9255_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys2,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_9256_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I4)
            <=> aa_nat_o(nth_o(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_9257_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_9258_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I4),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_int_o(aa_nat_fun_int_o(P,I4),aa_nat_int(nth_int(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_9259_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),nth_VEBT_VEBT(Xs3,I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_9260_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I4),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_o_o(aa_nat_fun_o_o(P,I4),aa_nat_o(nth_o(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_9261_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I4),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I4),aa_nat_nat(nth_nat(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_9262_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_9263_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys2,I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_9264_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
     => ( ! [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(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_9265_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
     => ( ! [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(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_9266_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S3),Xa) = vEBT_Node(Info,zero_zero_nat,Ts,S3) ).

% vebt_insert.simps(2)
tff(fact_9267_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_9268_neg__numeral__class_Osub__def,axiom,
    ! [K: num,L: num] : aa_num_rat(neg_numeral_sub_rat(K),L) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% neg_numeral_class.sub_def
tff(fact_9269_neg__numeral__class_Osub__def,axiom,
    ! [K: num,L: num] : aa_num_real(neg_numeral_sub_real(K),L) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,K)),aa_num_real(numeral_numeral_real,L)) ).

% neg_numeral_class.sub_def
tff(fact_9270_neg__numeral__class_Osub__def,axiom,
    ! [K: num,L: num] : aa_num_int(neg_numeral_sub_int(K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L)) ).

% neg_numeral_class.sub_def
tff(fact_9271_neg__numeral__class_Osub__def,axiom,
    ! [K: num,L: num] : neg_nu5755505904847501662nteger(K,L) = minus_8373710615458151222nteger(numera6620942414471956472nteger(K),numera6620942414471956472nteger(L)) ).

% neg_numeral_class.sub_def
tff(fact_9272_nth__mem,axiom,
    ! [Na: nat,Xs: list_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3941691890525107288d_enat(Xs))
     => aa_set3337841544853333262enat_o(member_Extended_enat(nth_Extended_enat(Xs,Na)),set_Extended_enat2(Xs)) ) ).

% nth_mem
tff(fact_9273_nth__mem,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Na)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_9274_nth__mem,axiom,
    ! [Na: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(nth_set_nat(Xs,Na)),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_9275_nth__mem,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Na)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_9276_nth__mem,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Na),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_9277_nth__mem,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => aa_set_o_o(member_o(aa_nat_o(nth_o(Xs),Na)),set_o2(Xs)) ) ).

% nth_mem
tff(fact_9278_nth__mem,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Na)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_9279_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),set_int2(Xs))
           => aa_int_o(P,X2) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_9280_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X2) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% list_ball_nth
tff(fact_9281_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),set_o2(Xs))
           => aa_o_o(P,(X2)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_9282_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => aa_nat_o(P,X2) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_9283_in__set__conv__nth,axiom,
    ! [Xa: extended_enat,Xs: list_Extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),set_Extended_enat2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s3941691890525107288d_enat(Xs))
          & ( nth_Extended_enat(Xs,I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_9284_in__set__conv__nth,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),set_real2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_9285_in__set__conv__nth,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_9286_in__set__conv__nth,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),set_int2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_9287_in__set__conv__nth,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_9288_in__set__conv__nth,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),set_o2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I4)
          <=> (Xa) ) ) ) ).

% in_set_conv_nth
tff(fact_9289_in__set__conv__nth,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),set_nat2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_9290_all__nth__imp__all__set,axiom,
    ! [Xs: list_Extended_enat,P: fun_Extended_enat_o,Xa: extended_enat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3941691890525107288d_enat(Xs))
         => aa_Extended_enat_o(P,nth_Extended_enat(Xs,I2)) )
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),set_Extended_enat2(Xs))
       => aa_Extended_enat_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_9291_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xa: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) )
     => ( aa_set_real_o(member_real(Xa),set_real2(Xs))
       => aa_real_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_9292_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Xa: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,nth_set_nat(Xs,I2)) )
     => ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
       => aa_set_nat_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_9293_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xa: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) )
     => ( aa_set_int_o(member_int(Xa),set_int2(Xs))
       => aa_int_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_9294_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xa: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) )
     => ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_9295_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xa: $o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) )
     => ( aa_set_o_o(member_o((Xa)),set_o2(Xs))
       => aa_o_o(P,(Xa)) ) ) ).

% all_nth_imp_all_set
tff(fact_9296_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xa: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat(Xa),set_nat2(Xs))
       => aa_nat_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_9297_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_int_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_9298_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_9299_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),set_o2(Xs))
         => aa_o_o(P,(X)) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_9300_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_nat_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_9301_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(bit_se545348938243370406it_int(M2,K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9302_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3) ).

% vebt_insert.simps(3)
tff(fact_9303_sub__non__negative,axiom,
    ! [Na: num,M2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(neg_numeral_sub_real(Na),M2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% sub_non_negative
tff(fact_9304_sub__non__negative,axiom,
    ! [Na: num,M2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(neg_numeral_sub_rat(Na),M2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% sub_non_negative
tff(fact_9305_sub__non__negative,axiom,
    ! [Na: num,M2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(neg_numeral_sub_int(Na),M2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na) ) ).

% sub_non_negative
tff(fact_9306_sub__non__positive,axiom,
    ! [Na: num,M2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(neg_numeral_sub_real(Na),M2)),zero_zero_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% sub_non_positive
tff(fact_9307_sub__non__positive,axiom,
    ! [Na: num,M2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(neg_numeral_sub_rat(Na),M2)),zero_zero_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% sub_non_positive
tff(fact_9308_sub__non__positive,axiom,
    ! [Na: num,M2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(neg_numeral_sub_int(Na),M2)),zero_zero_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M2) ) ).

% sub_non_positive
tff(fact_9309_sub__positive,axiom,
    ! [Na: num,M2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(neg_numeral_sub_real(Na),M2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% sub_positive
tff(fact_9310_sub__positive,axiom,
    ! [Na: num,M2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(neg_numeral_sub_rat(Na),M2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% sub_positive
tff(fact_9311_sub__positive,axiom,
    ! [Na: num,M2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(neg_numeral_sub_int(Na),M2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na) ) ).

% sub_positive
tff(fact_9312_sub__negative,axiom,
    ! [Na: num,M2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(neg_numeral_sub_real(Na),M2)),zero_zero_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% sub_negative
tff(fact_9313_sub__negative,axiom,
    ! [Na: num,M2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(neg_numeral_sub_rat(Na),M2)),zero_zero_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% sub_negative
tff(fact_9314_sub__negative,axiom,
    ! [Na: num,M2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(neg_numeral_sub_int(Na),M2)),zero_zero_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M2) ) ).

% sub_negative
tff(fact_9315_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),Na) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_9316_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(rotate1_VEBT_VEBT(Xs),Na) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_s6755466524823107622T_VEBT(Xs))) ) ) ).

% nth_rotate1
tff(fact_9317_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(rotate1_o(Xs)),Na)
      <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_size_list_o(Xs))) ) ) ).

% nth_rotate1
tff(fact_9318_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),Na) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_size_list_nat(Xs))) ) ) ).

% nth_rotate1
tff(fact_9319_in__children__def,axiom,
    ! [Na: nat,TreeList: list_VEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Na,TreeList),Xa)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Na)) ) ).

% in_children_def
tff(fact_9320_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_9321_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_9322_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_9323_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_9324_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)
     => ( ( Mi = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
             => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_9325_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_9326_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_9327_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Xa)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_9328_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ( Xa != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy)
           => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_9329_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_9330_subrelI,axiom,
    ! [R2: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ! [X2: code_integer,Y: code_integer] :
          ( member157494554546826820nteger(produc1086072967326762835nteger(X2,Y),R2)
         => member157494554546826820nteger(produc1086072967326762835nteger(X2,Y),S3) )
     => ord_le3725938330318615451nteger(R2,S3) ) ).

% subrelI
tff(fact_9331_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X2: code_integer,Y: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X2,(Y)),R2)
         => member1379723562493234055eger_o(produc6677183202524767010eger_o(X2,(Y)),S3) )
     => ord_le8980329558974975238eger_o(R2,S3) ) ).

% subrelI
tff(fact_9332_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
          ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Y),R2)
         => member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Y),S3) )
     => ord_le3000389064537975527at_nat(R2,S3) ) ).

% subrelI
tff(fact_9333_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y),S3) )
     => ord_le3146513528884898305at_nat(R2,S3) ) ).

% subrelI
tff(fact_9334_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X2,Y),R2)
         => member5262025264175285858nt_int(product_Pair_int_int(X2,Y),S3) )
     => ord_le2843351958646193337nt_int(R2,S3) ) ).

% subrelI
tff(fact_9335_ssubst__Pair__rhs,axiom,
    ! [R2: code_integer,S3: code_integer,R: set_Pr4811707699266497531nteger,S5: code_integer] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(R2,S3),R)
     => ( ( S5 = S3 )
       => member157494554546826820nteger(produc1086072967326762835nteger(R2,S5),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_9336_ssubst__Pair__rhs,axiom,
    ! [R2: code_integer,S3: $o,R: set_Pr448751882837621926eger_o,S5: $o] :
      ( member1379723562493234055eger_o(produc6677183202524767010eger_o(R2,(S3)),R)
     => ( ( (S5)
        <=> (S3) )
       => member1379723562493234055eger_o(produc6677183202524767010eger_o(R2,(S5)),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_9337_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S3: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S5: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(R2,S3),R)
     => ( ( S5 = S3 )
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(R2,S5),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_9338_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R: set_Pr1261947904930325089at_nat,S5: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S3),R)
     => ( ( S5 = S3 )
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S5),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_9339_ssubst__Pair__rhs,axiom,
    ! [R2: int,S3: int,R: set_Pr958786334691620121nt_int,S5: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(R2,S3),R)
     => ( ( S5 = S3 )
       => member5262025264175285858nt_int(product_Pair_int_int(R2,S5),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_9340_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_9341_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Ya: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
      <=> (Ya) )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ~ (Ya) )
       => ( ( ? [Uv2: $o] : Xa = vEBT_Leaf($true,(Uv2))
           => (Ya) )
         => ( ( ? [Uu2: $o] : Xa = vEBT_Leaf((Uu2),$true)
             => (Ya) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy)
               => ~ (Ya) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc)
                 => (Ya) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_9342_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_9343_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xa) ).

% vebt_member.simps(2)
tff(fact_9344_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy2)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_9345_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2)),Xa) ).

% vebt_member.simps(3)
tff(fact_9346_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_9347_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)),Uz2) ).

% VEBT_internal.membermima.simps(2)
tff(fact_9348_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( ( Xa != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_9349_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)),Xa) ).

% vebt_member.simps(4)
tff(fact_9350_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A12),A23)
    <=> ( ( ? [A7: $o,B6: $o] : A12 = vEBT_Leaf((A7),(B6))
          & ( A23 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X),N) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N)
            & ( 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))),N) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X),N) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N))
            & ( 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,N)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X),N) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N)
            & ( 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))),N) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N) )
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
            & 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))),A23))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I4 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X),N) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N))
            & ( 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,N)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) )
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
            & 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))),A23))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I4 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_9351_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A12),A23)
     => ( ( ? [A3: $o,B4: $o] : A12 = vEBT_Leaf((A3),(B4))
         => ( A23 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
             => ( ( A23 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary3),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 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_13)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
               => ( ( A23 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary3),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,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_13)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                  ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary3),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 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                             => ( ! [I3: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I3) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I3: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N2) = I3 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2)) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I3 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                    ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary3),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,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                               => ( ! [I3: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I3) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I3: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N2) = I3 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2)) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I3 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_9352_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2) )
             => ( ! [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))),M2))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [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))),M2))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Na)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Na) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_9353_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M2) )
             => ( ! [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))),M2))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [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))),M2))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Na)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Na) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_9354_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_9355_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,Xa: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_9356_divmod__step__eq,axiom,
    ! [L: num,Q5: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q5),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),aa_nat7603221163087113075at_nat(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))),Q5)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(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))),Q5)),R2)) ).

% divmod_step_eq
tff(fact_9357_divmod__step__eq,axiom,
    ! [L: num,Q5: int,R2: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q5,R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q5)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q5),R2)) ).

% divmod_step_eq
tff(fact_9358_divmod__step__eq,axiom,
    ! [L: num,Q5: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q5,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q5),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q5),R2)) ).

% divmod_step_eq
tff(fact_9359_divides__aux__eq,axiom,
    ! [Q5: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q5,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_9360_divides__aux__eq,axiom,
    ! [Q5: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q5),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_9361_divides__aux__eq,axiom,
    ! [Q5: int,R2: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q5,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_9362_product__nth,axiom,
    ! [Na: nat,Xs: list_Code_integer,Ys2: list_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys2)))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys2),Na) = produc1086072967326762835nteger(nth_Code_integer(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s3445333598471063425nteger(Ys2))),nth_Code_integer(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s3445333598471063425nteger(Ys2)))) ) ) ).

% product_nth
tff(fact_9363_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),Na) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_9364_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys2)))
     => ( nth_Pr3474266648193625910T_VEBT(produc662631939642741121T_VEBT(Xs,Ys2),Na) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_9365_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr7514405829937366042_int_o(product_int_o(Xs,Ys2),Na) = product_Pair_int_o(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_9366_product__nth,axiom,
    ! [Na: nat,Xs: list_Code_integer,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys2),Na) = produc6677183202524767010eger_o(nth_Code_integer(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_9367_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys2)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys2),Na) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_9368_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys2),Na) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_9369_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),Na) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_9370_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys2),Na) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_9371_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys2)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),Na) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_9372_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_9373_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xa)) = Xs ) ) ).

% list_update_beyond
tff(fact_9374_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_9375_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_9376_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xa),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_9377_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xa: $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,(Xa))),I)
      <=> (Xa) ) ) ).

% nth_list_update_eq
tff(fact_9378_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xa: 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,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_9379_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_9380_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_9381_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_9382_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: 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,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_9383_set__update__subsetI,axiom,
    ! [Xs: list_Extended_enat,A2: set_Extended_enat,Xa: extended_enat,I: nat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
       => aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(list_u3071683517702156500d_enat(Xs,I,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_9384_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,Xa: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_9385_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,Xa: set_nat,I: nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_9386_set__update__subsetI,axiom,
    ! [Xs: list_o,A2: set_o,Xa: $o,I: nat] :
      ( aa_set_o_o(ord_less_eq_set_o(set_o2(Xs)),A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,I,(Xa)))),A2) ) ) ).

% set_update_subsetI
tff(fact_9387_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,Xa: nat,I: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_9388_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,Xa: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( member_VEBT_VEBT(Xa,A2)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xa)),A2) ) ) ).

% set_update_subsetI
tff(fact_9389_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,Xa: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_9390_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D5: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D5)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_9391_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,Xa: $o] : aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,I,(Xa)))),insert_o((Xa),set_o2(Xs))) ).

% set_update_subset_insert
tff(fact_9392_set__update__subset__insert,axiom,
    ! [Xs: list_set_nat,I: nat,Xa: set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,Xa))),insert_set_nat(Xa,set_set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_9393_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,Xa: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xa))),insert_real(Xa,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_9394_set__update__subset__insert,axiom,
    ! [Xs: list_Extended_enat,I: nat,Xa: extended_enat] : aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(list_u3071683517702156500d_enat(Xs,I,Xa))),insert_Extended_enat(Xa,set_Extended_enat2(Xs))) ).

% set_update_subset_insert
tff(fact_9395_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,Xa: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,Xa))),insert_nat(Xa,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_9396_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xa)),insert_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_9397_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,Xa: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xa))),insert_int(Xa,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_9398_set__update__memI,axiom,
    ! [Na: nat,Xs: list_Extended_enat,Xa: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3941691890525107288d_enat(Xs))
     => aa_set3337841544853333262enat_o(member_Extended_enat(Xa),set_Extended_enat2(list_u3071683517702156500d_enat(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_9399_set__update__memI,axiom,
    ! [Na: nat,Xs: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xa),set_real2(list_update_real(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_9400_set__update__memI,axiom,
    ! [Na: nat,Xs: list_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_9401_set__update__memI,axiom,
    ! [Na: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xa),set_int2(list_update_int(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_9402_set__update__memI,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(Xa,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_9403_set__update__memI,axiom,
    ! [Na: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => aa_set_o_o(member_o((Xa)),set_o2(list_update_o(Xs,Na,(Xa)))) ) ).

% set_update_memI
tff(fact_9404_set__update__memI,axiom,
    ! [Na: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xa),set_nat2(list_update_nat(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_9405_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,Xa) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_9406_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xa) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_9407_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xa)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xa) ) ) ) ).

% list_update_same_conv
tff(fact_9408_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,Xa) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_9409_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xa: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xa)),J) = $ite(I = J,Xa,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_9410_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xa),J) = $ite(I = J,Xa,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_9411_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xa: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xa))),J)
      <=> $ite(I = J,(Xa),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_9412_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xa: nat,J: 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,Xa)),J) = $ite(I = J,Xa,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_9413_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B4: $o,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),X2)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_9414_vebt__insert_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B4: $o,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),X2)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S4),X2)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4),X2)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3),X2)
           => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3),X2) ) ) ) ) ).

% vebt_insert.cases
tff(fact_9415_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy),Uz)
       => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb),X2)
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc),X2)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_9416_vebt__member_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B4: $o,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),X2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X2)
       => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz),X2)
         => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc),X2)
           => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3),X2) ) ) ) ) ).

% vebt_member.cases
tff(fact_9417_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))),unique5026877609467782581ep_nat(aa_num_num(bit1,Na),unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_9418_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),unique5024387138958732305ep_int(aa_num_num(bit1,Na),unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_9419_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M2))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_9420_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M2))),unique5026877609467782581ep_nat(aa_num_num(bit1,Na),unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_9421_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),unique5024387138958732305ep_int(aa_num_num(bit1,Na),unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_9422_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M2),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M2))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_9423_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q5: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
     => ( eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B2,product_Pair_int_int(Q5,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2),product_Pair_int_int(Q5,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_9424_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q5: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( eucl_rel_int(A,B2,product_Pair_int_int(Q5,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2),product_Pair_int_int(Q5,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_9425_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_9426_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_9427_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique5052692396658037445od_int(M2,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M2),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_9428_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique3479559517661332726nteger(M2,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M2),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_9429_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique5055182867167087721od_nat(M2,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M2)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_9430_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Na)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_9431_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_9432_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_9433_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Na)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_9434_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_9435_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_9436_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q5: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q5,R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q5)),R2) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q5 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_9437_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q5: int] :
      ( ( sgn_sgn_int(R2) = sgn_sgn_int(L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L)),R2) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q5,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9438_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A12,A23,A32)
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L3: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L3 )
            & ( A32 = product_Pair_int_int(Q6,zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L3 )
            & ( A32 = product_Pair_int_int(Q6,R5) )
            & ( sgn_sgn_int(R5) = sgn_sgn_int(L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9439_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A12,A23,A32)
     => ( ( ( A23 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A12) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A23 != zero_zero_int )
               => ( A12 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A23) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R3) )
               => ( ( sgn_sgn_int(R3) = sgn_sgn_int(A23) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A23))
                   => ( A12 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A23)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9440_divmod__divmod__step,axiom,
    ! [M2: num,Na: num] :
      unique5055182867167087721od_nat(M2,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M2)),unique5026877609467782581ep_nat(Na,unique5055182867167087721od_nat(M2,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_9441_divmod__divmod__step,axiom,
    ! [M2: num,Na: num] :
      unique5052692396658037445od_int(M2,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M2)),unique5024387138958732305ep_int(Na,unique5052692396658037445od_int(M2,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_9442_divmod__divmod__step,axiom,
    ! [M2: num,Na: num] :
      unique3479559517661332726nteger(M2,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M2),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M2)),unique4921790084139445826nteger(Na,unique3479559517661332726nteger(M2,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_9443_option_Osize_I4_J,axiom,
    ! [X23: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_9444_option_Osize_I4_J,axiom,
    ! [X23: num] : size_size_option_num(some_num(X23)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_9445_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_Pr2602117396471556196at_nat,X23: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xa,some_P7363390416028606310at_nat(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xa,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_9446_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_num_nat,X23: num] : size_option_num(Xa,some_num(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xa,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_9447_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Ya )
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => ( Ya != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B4)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B4))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,zero_zero_nat,Ts2,S4) )
             => ( Ya != vEBT_Node(Info2,zero_zero_nat,Ts2,S4) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) )
               => ( Ya != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) )
                 => ( Ya != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) ) )
             => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) )
                   => ( Ya != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),Mi3,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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi3 )
                                  | ( Xaa = Ma3 ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),Xaa,Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma3))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary3,h),Summary3)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_9448_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
      <=> (Ya) )
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => ( (Ya)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Ya) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz)
             => (Ya) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)
               => (Ya) )
             => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3)
                   => ( (Ya)
                    <=> ~ $ite(
                            Xaa = Mi3,
                            $true,
                            $ite(
                              Xaa = Ma3,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_9449_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz)
           => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)
             => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3)
                   => $ite(
                        Xaa = Mi3,
                        $true,
                        $ite(
                          Xaa = Ma3,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_9450_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Xa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_9451_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_9452_finite__Collect__disjI,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ag(P),Q)))
    <=> ( aa_set_real_o(finite_finite_real,collect_real(P))
        & aa_set_real_o(finite_finite_real,collect_real(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9453_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ah(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9454_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ai(P),Q)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9455_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aj(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9456_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ak(P),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9457_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_al(P),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9458_finite__Collect__disjI,axiom,
    ! [P: fun_Extended_enat_o,Q: fun_Extended_enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3141604562678283816enat_o(aTP_Lamm_am(P),Q)))
    <=> ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P))
        & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_9459_finite__Collect__conjI,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ( aa_set_real_o(finite_finite_real,collect_real(P))
        | aa_set_real_o(finite_finite_real,collect_real(Q)) )
     => aa_set_real_o(finite_finite_real,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_an(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9460_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9461_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9462_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9463_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P))
        | aa_set_int_o(finite_finite_int,collect_int(Q)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9464_finite__Collect__conjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_as(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9465_finite__Collect__conjI,axiom,
    ! [P: fun_Extended_enat_o,Q: fun_Extended_enat_o] :
      ( ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P))
        | aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(Q)) )
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3141604562678283816enat_o(aTP_Lamm_at(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_9466_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_au,A2))) ) ).

% finite_Collect_subsets
tff(fact_9467_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_av(A2))) ) ).

% finite_Collect_subsets
tff(fact_9468_finite__Collect__subsets,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => finite5468666774076196335d_enat(collec2260605976452661553d_enat(aTP_Lamm_aw(A2))) ) ).

% finite_Collect_subsets
tff(fact_9469_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_ax(A2))) ) ).

% finite_Collect_subsets
tff(fact_9470_finite__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ay(Na),C2))) ) ).

% finite_nth_roots
tff(fact_9471_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_az(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_9472_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ba,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_9473_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_bb,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_9474_singleton__conv,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(aTP_Lamm_bc,(A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv
tff(fact_9475_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bd,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_9476_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_be,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_9477_singleton__conv,axiom,
    ! [A: extended_enat] : collec4429806609662206161d_enat(aa_Ext6368937152127170207enat_o(aTP_Lamm_bf,A)) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ).

% singleton_conv
tff(fact_9478_singleton__conv2,axiom,
    ! [A: list_nat] : collect_list_nat(fequal_list_nat(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_9479_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_9480_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_9481_singleton__conv2,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(fequal_o,(A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv2
tff(fact_9482_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set7861427066675100071_nat_o(fequal_set_nat,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_9483_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_9484_singleton__conv2,axiom,
    ! [A: extended_enat] : collec4429806609662206161d_enat(aa_Ext6368937152127170207enat_o(fequal_Extended_enat,A)) = insert_Extended_enat(A,bot_bo7653980558646680370d_enat) ).

% singleton_conv2
tff(fact_9485_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,K))) ).

% finite_Collect_less_nat
tff(fact_9486_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bh,K))) ).

% finite_Collect_le_nat
tff(fact_9487_card__Collect__less__nat,axiom,
    ! [Na: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Na))) = Na ).

% card_Collect_less_nat
tff(fact_9488_finite__interval__int1,axiom,
    ! [A: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bi(A),B2))) ).

% finite_interval_int1
tff(fact_9489_finite__interval__int4,axiom,
    ! [A: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bj(A),B2))) ).

% finite_interval_int4
tff(fact_9490_card__Collect__le__nat,axiom,
    ! [Na: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bh,Na))) = aa_nat_nat(suc,Na) ).

% card_Collect_le_nat
tff(fact_9491_finite__interval__int3,axiom,
    ! [A: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bk(A),B2))) ).

% finite_interval_int3
tff(fact_9492_finite__interval__int2,axiom,
    ! [A: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bl(A),B2))) ).

% finite_interval_int2
tff(fact_9493_pred__subset__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S: set_Pr4811707699266497531nteger] :
      ( ord_le6737357399843569584eger_o(aTP_Lamm_bm(R),aTP_Lamm_bm(S))
    <=> ord_le3725938330318615451nteger(R,S) ) ).

% pred_subset_eq2
tff(fact_9494_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_bn(R),aTP_Lamm_bn(S))
    <=> ord_le8980329558974975238eger_o(R,S) ) ).

% pred_subset_eq2
tff(fact_9495_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ord_le3650406611588909774_nat_o(aTP_Lamm_bo(R),aTP_Lamm_bo(S))
    <=> ord_le3000389064537975527at_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_9496_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_bp(R),aTP_Lamm_bp(S))
    <=> ord_le3146513528884898305at_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_9497_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_bq(R),aTP_Lamm_bq(S))
    <=> ord_le2843351958646193337nt_int(R,S) ) ).

% pred_subset_eq2
tff(fact_9498_inf__Int__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S: set_Pr4811707699266497531nteger,X4: code_integer,Xa2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(inf_in115219751047318782eger_o(aTP_Lamm_bm(R),aTP_Lamm_bm(S)),X4),Xa2)
    <=> member157494554546826820nteger(produc1086072967326762835nteger(X4,Xa2),inf_in8876636753061821161nteger(R,S)) ) ).

% inf_Int_eq2
tff(fact_9499_inf__Int__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S: set_Pr448751882837621926eger_o,X4: code_integer,Xa2: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(inf_in2596584417404321711un_o_o(aTP_Lamm_bn(R),aTP_Lamm_bn(S)),X4),(Xa2))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Xa2)),inf_in2046216803622501944eger_o(R,S)) ) ).

% inf_Int_eq2
tff(fact_9500_inf__Int__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat,X4: product_prod_nat_nat,Xa2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(inf_in8070068280171780892_nat_o(aTP_Lamm_bo(R),aTP_Lamm_bo(S)),X4),Xa2)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X4,Xa2),inf_in4302113700860409141at_nat(R,S)) ) ).

% inf_Int_eq2
tff(fact_9501_inf__Int__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,X4: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(inf_in8908015913983109984_int_o(aTP_Lamm_bq(R),aTP_Lamm_bq(S)),X4),Xa2)
    <=> member5262025264175285858nt_int(product_Pair_int_int(X4,Xa2),inf_in2269163501485487111nt_int(R,S)) ) ).

% inf_Int_eq2
tff(fact_9502_inf__Int__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X4: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aTP_Lamm_bp(R),aTP_Lamm_bp(S)),X4),Xa2)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,R),S)) ) ).

% inf_Int_eq2
tff(fact_9503_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_9504_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Xa3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X,Xa3),R)
        <=> member5262025264175285858nt_int(product_Pair_int_int(X,Xa3),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_9505_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( finite_card_complex(collect_complex(aTP_Lamm_br(Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_9506_card__nth__roots,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_bs(C2),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_9507_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_bt(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_9508_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(F,N2))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu(F),U))) ) ).

% finite_less_ub
tff(fact_9509_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_9510_nat__less__as__int,axiom,
    ! [X4: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_less_as_int
tff(fact_9511_nat__leq__as__int,axiom,
    ! [X4: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_leq_as_int
tff(fact_9512_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bv,M2))) ) ).

% finite_divisors_nat
tff(fact_9513_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M5)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bw(M5),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9514_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M5)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bx(M5),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bw(M5),I))) ) ) ).

% card_less_Suc
tff(fact_9515_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M5)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bx(M5),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bw(M5),I))) ) ) ).

% card_less_Suc2
tff(fact_9516_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_9517_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S3)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_9518_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_9519_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_9520_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_9521_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
      <=> (Ya) )
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => ( (Ya)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Ya) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S4: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4)
               => ( (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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_9522_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S4: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4)
             => ~ $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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_9523_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S4: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4)
               => $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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_9524_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb)
           => ~ ( ( Xaa = Mi3 )
                | ( Xaa = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc)
             => ~ ( ( Xaa = Mi3 )
                  | ( Xaa = Ma3 )
                  | $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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : Xa = 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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_9525_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary3: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3)
             => ~ $ite(
                    Xaa = Mi3,
                    $true,
                    $ite(
                      Xaa = Ma3,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_9526_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xa != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy)
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb)
               => ( ( Xaa = Mi3 )
                  | ( Xaa = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc)
                 => ( ( Xaa = Mi3 )
                    | ( Xaa = Ma3 )
                    | $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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xa = 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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_9527_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
      <=> (Ya) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => (Ya) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy)
           => (Ya) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb)
               => ( (Ya)
                <=> ~ ( ( Xaa = Mi3 )
                      | ( Xaa = Ma3 ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc)
                 => ( (Ya)
                  <=> ~ ( ( Xaa = Mi3 )
                        | ( Xaa = Ma3 )
                        | $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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xa = 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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_9528_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_by,Xa)) ) ).

% monoseq_arctan_series
tff(fact_9529_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( ( Ya = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B4)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B4))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero_nat,Ts2,S4) )
               => ( ( Ya = vEBT_Node(Info2,zero_zero_nat,Ts2,S4) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S4),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) )
                 => ( ( Ya = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) )
                   => ( ( Ya = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3),Xaa)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) )
                     => ( ( Ya = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),Mi3,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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi3 )
                                      | ( Xaa = Ma3 ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),Xaa,Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma3))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary3,h),Summary3)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_9530_ln__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_bz,Xa)) ) ) ) ).

% ln_series
tff(fact_9531_arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arctan,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ca,Xa)) ) ) ).

% arctan_series
tff(fact_9532_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xa)) ) ) ).

% monoseq_realpow
tff(fact_9533_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( ( (Ya)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B4),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Ya)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz) )
                 => ( ~ (Ya)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xa = 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(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc),Xaa)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) )
                     => ( ( (Ya)
                        <=> $ite(
                              Xaa = Mi3,
                              $true,
                              $ite(
                                Xaa = Ma3,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_9534_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc),Xaa)) )
               => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3),Xaa))
                       => $ite(
                            Xaa = Mi3,
                            $true,
                            $ite(
                              Xaa = Ma3,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_9535_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( ( (Ya)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B4),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Ya)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4) )
                 => ( ( (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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_9536_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_9537_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2,S4),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_9538_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B4)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary3),Xaa))
                 => ~ $ite(
                        Xaa = Mi3,
                        $true,
                        $ite(
                          Xaa = Ma3,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi3),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_9539_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy),Xaa)) )
           => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb),Xaa))
                   => ( ( Xaa = Mi3 )
                      | ( Xaa = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc),Xaa))
                     => ( ( Xaa = Mi3 )
                        | ( Xaa = Ma3 )
                        | $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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_9540_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Ya)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy) )
               => ( ~ (Ya)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy),Xaa)) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb) )
                 => ( ( (Ya)
                    <=> ( ( Xaa = Mi3 )
                        | ( Xaa = Ma3 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb),Xaa)) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc) )
                   => ( ( (Ya)
                      <=> ( ( Xaa = Mi3 )
                          | ( Xaa = Ma3 )
                          | $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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc),Xaa)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xa = 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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_9541_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va2,Vb),Xaa))
               => ~ ( ( Xaa = Mi3 )
                    | ( Xaa = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc),Xaa))
                 => ~ ( ( Xaa = Mi3 )
                      | ( Xaa = Ma3 )
                      | $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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_9542_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ca,Xa)) ) ).

% summable_arctan_series
tff(fact_9543_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Ya )
     => ( ( ( Xa = zero_zero_nat )
         => ( Ya != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( 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,Va))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_9544_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G3: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G3,N2)) )
     => ( summable_real(G3)
       => summable_real(aTP_Lamm_cb(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_9545_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_cb(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_cb(F))) ) ).

% summable_rabs
tff(fact_9546_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_cc(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_9547_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_nat_nat(suc,zero_zero_nat)) = 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_bg,Na))) ).

% mask_eq_sum_exp_nat
tff(fact_9548_gauss__sum__nat,axiom,
    ! [Na: nat] : groups3542108847815614940at_nat(aTP_Lamm_cd,set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_9549_arith__series__nat,axiom,
    ! [A: nat,D: nat,Na: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ce(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_9550_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va3: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = $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,Va3))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_9551_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Ya = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( 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,Va))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_9552_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xa)
            & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9553_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Xa)
          & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9554_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9555_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_9556_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_9557_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_9558_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_9559_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ? [K2: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K2)) ) ).

% finite_nat_bounded
tff(fact_9560_finite__nat__iff__bounded,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_9561_sum__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_cg,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ay(Na),C2))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_9562_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_cg,collect_complex(aTP_Lamm_br(Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_9563_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B3: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ch(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_9564_Maclaurin__exp__le,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ci,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_9565_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xa)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xa)),Na))) ).

% Maclaurin_sin_bound
tff(fact_9566_Sum__Icc__int,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),Na)
     => ( groups4538972089207619220nt_int(aTP_Lamm_cj,set_or1266510415728281911st_int(M2,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M2),aa_int_int(aa_int_fun_int_int(minus_minus_int,M2),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_9567_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [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(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_9568_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Na: nat] :
      ( ( Xa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ci,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9569_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xa)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ck,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9570_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ck,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9571_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ck,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T4),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,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9572_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_9573_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_9574_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_9575_atMost__atLeast0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,Na) = set_or1269000886237332187st_nat(zero_zero_nat,Na) ).

% atMost_atLeast0
tff(fact_9576_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_9577_finite__nat__iff__bounded__le,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_9578_sum__choose__diagonal,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cl(M2),Na),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),M2) ) ) ).

% sum_choose_diagonal
tff(fact_9579_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Na)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9580_polynomial__product__nat,axiom,
    ! [M2: nat,A: fun_nat_nat,Na: nat,B2: fun_nat_nat,Xa: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_nat(B2,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cm(A),Xa),aa_nat_set_nat(set_ord_atMost_nat,M2))),groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cm(B2),Xa),aa_nat_set_nat(set_ord_atMost_nat,Na))) = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_co(A),B2),Xa),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_9581_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A12))
     => ( ! [I2: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I2,J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A12) ) ) ).

% upto.pinduct
tff(fact_9582_arctan__def,axiom,
    ! [Ya: real] : aa_real_real(arctan,Ya) = the_real(aa_real_fun_real_o(aTP_Lamm_cp,Ya)) ).

% arctan_def
tff(fact_9583_arcsin__def,axiom,
    ! [Ya: real] : aa_real_real(arcsin,Ya) = the_real(aa_real_fun_real_o(aTP_Lamm_cq,Ya)) ).

% arcsin_def
tff(fact_9584_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xa) = the_real(aTP_Lamm_cr) ) ) ).

% ln_neg_is_const
tff(fact_9585_arccos__def,axiom,
    ! [Ya: real] : aa_real_real(arccos,Ya) = the_real(aa_real_fun_real_o(aTP_Lamm_cs,Ya)) ).

% arccos_def
tff(fact_9586_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_ct) ).

% pi_half
tff(fact_9587_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_ct)) ).

% pi_def
tff(fact_9588_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Ya: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
      <=> (Ya) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( (Ya)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Ya)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xa = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Ya)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy) )
                   => ( (Ya)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy)) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc) )
                     => ( ~ (Ya)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9589_floor__real__def,axiom,
    ! [Xa: real] : aa_real_int(archim6058952711729229775r_real,Xa) = the_int(aTP_Lamm_cu(Xa)) ).

% floor_real_def
tff(fact_9590_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ! [Uv2: $o] :
              ( ( Xa = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xa = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9591_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9592_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_cv(L),Qr) ).

% divmod_step_nat_def
tff(fact_9593_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_cw(L),Qr) ).

% divmod_step_int_def
tff(fact_9594_divmod__nat__if,axiom,
    ! [M2: nat,Na: nat] :
      divmod_nat(M2,Na) = $ite(
        ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M2),
        produc2626176000494625587at_nat(aTP_Lamm_cx,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na),Na)) ) ).

% divmod_nat_if
tff(fact_9595_floor__rat__def,axiom,
    ! [Xa: rat] : aa_rat_int(archim3151403230148437115or_rat,Xa) = the_int(aTP_Lamm_cy(Xa)) ).

% floor_rat_def
tff(fact_9596_abs__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_rat_def
tff(fact_9597_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9598_less__eq__rat__def,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% less_eq_rat_def
tff(fact_9599_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S4)
           => ! [T4: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T4)
               => ( R2 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S4),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9600_fact__eq__fact__times,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),groups708209901874060359at_nat(aTP_Lamm_cd,set_or1269000886237332187st_nat(aa_nat_nat(suc,Na),M2))) ) ) ).

% fact_eq_fact_times
tff(fact_9601_fact__div__fact,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = groups708209901874060359at_nat(aTP_Lamm_cd,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),M2)) ) ) ).

% fact_div_fact
tff(fact_9602_normalize__negative,axiom,
    ! [Q5: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q5),zero_zero_int)
     => ( normalize(product_Pair_int_int(P3,Q5)) = normalize(product_Pair_int_int(uminus_uminus_int(P3),uminus_uminus_int(Q5))) ) ) ).

% normalize_negative
tff(fact_9603_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q5: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q5) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q5) ) ).

% normalize_denom_pos
tff(fact_9604_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9605_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9606_Least__eq__0,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9607_Least__Suc2,axiom,
    ! [P: fun_nat_o,Na: nat,Q: fun_nat_o,M2: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(Q,M2)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Q,K2) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9608_Least__Suc,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_cz(P))) ) ) ) ).

% Least_Suc
tff(fact_9609_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_da(D))) ).

% int_ge_less_than2_def
tff(fact_9610_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_db(D))) ).

% int_ge_less_than_def
tff(fact_9611_sum__power2,axiom,
    ! [K: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9612_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or4665077453230672383an_nat(L,U)) ).

% finite_atLeastLessThan
tff(fact_9613_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : set_or4665077453230672383an_nat(M2,aa_nat_nat(suc,M2)) = insert_nat(M2,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9614_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
          & aa_nat_o(P,M) )
    <=> ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or4665077453230672383an_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less_eq
tff(fact_9615_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => aa_nat_o(P,M) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_or4665077453230672383an_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less_eq
tff(fact_9616_lessThan__atLeast0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Na) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ).

% lessThan_atLeast0
tff(fact_9617_atLeastLessThan0,axiom,
    ! [M2: nat] : set_or4665077453230672383an_nat(M2,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9618_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(Na,set_or4665077453230672383an_nat(zero_zero_nat,Na)) ).

% atLeast0_lessThan_Suc
tff(fact_9619_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9620_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2))))
     => ( A2 = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9621_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or4665077453230672383an_nat(I,J)),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9622_atLeastLessThanSuc,axiom,
    ! [M2: nat,Na: nat] :
      set_or4665077453230672383an_nat(M2,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na),insert_nat(Na,set_or4665077453230672383an_nat(M2,Na)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9623_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_9624_prod__Suc__fact,axiom,
    ! [Na: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_fact
tff(fact_9625_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N4)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9626_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3542108847815614940at_nat(aTP_Lamm_cd,set_or4665077453230672383an_nat(zero_zero_nat,finite_card_nat(S)))),groups3542108847815614940at_nat(aTP_Lamm_cd,S)) ).

% card_sum_le_nat_sum
tff(fact_9627_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Na) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Na)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9628_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A: fun_nat_nat,B2: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B2,J2)),aa_nat_nat(B2,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dc(A),B2),set_or4665077453230672383an_nat(zero_zero_nat,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3542108847815614940at_nat(A,set_or4665077453230672383an_nat(zero_zero_nat,Na))),groups3542108847815614940at_nat(B2,set_or4665077453230672383an_nat(zero_zero_nat,Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9629_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_dd(L),Qr) ).

% divmod_step_integer_def
tff(fact_9630_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),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9631_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_9632_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9633_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),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,code_integer_of_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(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9634_bezw__0,axiom,
    ! [Xa: nat] : bezw(Xa,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9635_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Ya )
     => ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9636_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),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).

% abs_integer_code
tff(fact_9637_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_9638_less__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,code_integer_of_int(Xa)),code_integer_of_int(Xb3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xb3) ) ).

% less_integer.abs_eq
tff(fact_9639_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,code_integer_of_int(Xa)),code_integer_of_int(Xb3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xb3) ) ).

% less_eq_integer.abs_eq
tff(fact_9640_binomial__def,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_de(Na),K))) ).

% binomial_def
tff(fact_9641_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M2: nat] :
      nat_prod_decode_aux(K,M2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M2)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9642_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Ya )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9643_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S)))
          & ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),finite_card_nat(S))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N3)),S) ) ) ) ).

% finite_enumerate
tff(fact_9644_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_df(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9645_Sup__nat__empty,axiom,
    complete_Sup_Sup_nat(bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9646_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(complete_Inf_Inf_nat(K4)),K4) ) ).

% Inf_nat_def1
tff(fact_9647_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_dg(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            produc6499014454317279255nteger(uminus1351360451143612070nteger,
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_dh(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9648_Suc__funpow,axiom,
    ! [Na: nat] : compow_fun_nat_nat(Na,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Na) ).

% Suc_funpow
tff(fact_9649_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Na))) ).

% eval_nat_numeral(2)
tff(fact_9650_one__plus__BitM,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Na)) = aa_num_num(bit0,Na) ).

% one_plus_BitM
tff(fact_9651_BitM__plus__one,axiom,
    ! [Na: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Na)),one) = aa_num_num(bit0,Na) ).

% BitM_plus_one
tff(fact_9652_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9653_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9654_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_di,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9655_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bh,aTP_Lamm_bg) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9656_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),
        uminus_uminus_int(code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_dj,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9657_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_9658_less__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(Xa)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_9659_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_9660_less__eq__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(Xa)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9661_rat__floor__lemma,axiom,
    ! [A: int,B2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))),aa_int_rat(fract(A),B2))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9662_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xa: nat,Ya: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dk,C2),set_or4665077453230672383an_nat(Xa,Ya)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),Ya),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),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,Xa),Ya),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9663_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_dl,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9664_less__rat,axiom,
    ! [B2: int,D: int,A: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),aa_int_rat(fract(C2),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ).

% less_rat
tff(fact_9665_le__rat,axiom,
    ! [B2: int,D: int,A: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B2)),aa_int_rat(fract(C2),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ).

% le_rat
tff(fact_9666_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9667_Rat__induct__pos,axiom,
    ! [P: fun_rat_o,Q5: rat] :
      ( ! [A3: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_rat_o(P,aa_int_rat(fract(A3),B4)) )
     => aa_rat_o(P,Q5) ) ).

% Rat_induct_pos
tff(fact_9668_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9669_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9670_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9671_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_9672_Fract__less__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9673_zero__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9674_Fract__less__one__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% Fract_less_one_iff
tff(fact_9675_one__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ) ).

% one_less_Fract_iff
tff(fact_9676_Fract__le__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B2)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9677_zero__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9678_Fract__le__one__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B2)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% Fract_le_one_iff
tff(fact_9679_one__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% one_le_Fract_iff
tff(fact_9680_positive__rat,axiom,
    ! [A: int,B2: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ).

% positive_rat
tff(fact_9681_less__rat__def,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Ya)
    <=> aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Ya),Xa)) ) ).

% less_rat_def
tff(fact_9682_finite__int__iff__bounded__le,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S)),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9683_finite__int__iff__bounded,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S)),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_9684_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_9685_suminf__eq__SUP__real,axiom,
    ! [X3: fun_nat_real] :
      ( summable_real(X3)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X3,I2))
       => ( suminf_real(X3) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_dm(X3),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9686_nat__not__finite,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% nat_not_finite
tff(fact_9687_infinite__UNIV__nat,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_nat
tff(fact_9688_range__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( image_nat_nat(infini8530281810654367211te_nat(S),top_top_set_nat) = S ) ) ).

% range_enumerate
tff(fact_9689_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_9690_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dn,Na),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ) ) ).

% range_mod
tff(fact_9691_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9692_root__def,axiom,
    ! [Na: nat,Xa: real] :
      aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_do,Na)),Xa)) ).

% root_def
tff(fact_9693_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9694_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xa: real,D2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( Xa != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
               => ( D2 = 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,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
               => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D2,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9695_DERIV__even__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9696_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_dp,suminf_real(aa_real_fun_nat_real(aTP_Lamm_dq,Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9697_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B2: real,G3: fun_real_real,G4: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(A,B2))
         => has_fi5821293074295781190e_real(G3,aa_real_real(G4,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(A,B2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G4,X2)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G3,A)),aa_real_real(G3,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9698_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9699_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B2)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9700_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9701_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9702_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9703_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9704_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B2)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9705_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9706_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9707_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9708_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H3))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9709_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9710_MVT2,axiom,
    ! [A: real,B2: real,F: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),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(F,B2)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),aa_real_real(F4,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9711_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [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,Xa),Y))),D)
             => ( aa_real_real(F,Xa) = aa_real_real(F,Y) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9712_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9713_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [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,Xa),Y))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xa)),aa_real_real(F,Y)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9714_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [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,Xa),Y))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y)),aa_real_real(F,Xa)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9715_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9716_DERIV__pow,axiom,
    ! [Na: nat,Xa: real,S3: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_dr,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xa,S3)) ).

% DERIV_pow
tff(fact_9717_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ds,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9718_DERIV__log,axiom,
    ! [Xa: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => 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)),Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9719_DERIV__fun__powr,axiom,
    ! [G3: fun_real_real,M2: real,Xa: real,R2: real] :
      ( has_fi5821293074295781190e_real(G3,M2,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G3,Xa))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_dt(G3),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G3,Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M2),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9720_DERIV__powr,axiom,
    ! [G3: fun_real_real,M2: real,Xa: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G3,M2,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G3,Xa))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xa,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_du(G3),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G3,Xa)),aa_real_real(F,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G3,Xa)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M2),aa_real_real(F,Xa))),aa_real_real(G3,Xa)))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9721_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9722_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D2: real] :
      ( ( Xa != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( D2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
           => ( D2 = 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,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D2,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9723_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xa,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9724_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xa,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9725_DERIV__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9726_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9727_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9728_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M4: nat,X2: 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)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
          & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_dv(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9729_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,X2: 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)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_dv(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9730_DERIV__odd__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( Xa != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9731_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_dw(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9732_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_dw(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9733_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_dw(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9734_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( Xa != zero_zero_real )
         => ( ! [M4: nat,X2: 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)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
                & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_dv(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9735_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_dv(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9736_Taylor__down,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B2)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C2)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_dx(Diff),A),C2),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9737_Taylor__up,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B2)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),B2)
                  & ( aa_real_real(F,B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_dx(Diff),B2),C2),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9738_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C2: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,A),Xa)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),B2)
                 => ( ( Xa != C2 )
                   => ? [T4: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),C2),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C2) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xa) ) )
                        & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_dy(Diff),C2),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),C2)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9739_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B: real] :
      ( ! [M4: nat,T4: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M3: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ea(Na),Diff),B),M3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_eb(Diff),M3),T5),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M3))))),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,T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M3)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M3))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9740_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_ec(F),X2)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_ee(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ec(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9741_DERIV__isconst3,axiom,
    ! [A: real,B2: real,Xa: real,Ya: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_set_real_o(member_real(Xa),set_or1633881224788618240n_real(A,B2))
       => ( aa_set_real_o(member_real(Ya),set_or1633881224788618240n_real(A,B2))
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(A,B2))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) )
           => ( aa_real_real(F,Xa) = aa_real_real(F,Ya) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9742_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,A: real,B2: real,L4: fun_nat_real] :
      ( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ef(F),N2),aa_nat_real(aa_real_fun_nat_real(F4,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(A,B2))
           => summable_real(aa_real_fun_nat_real(F,X2)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A,B2))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L4)
             => ( ! [N2: nat,X2: real,Y: real] :
                    ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(A,B2))
                   => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,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(F,X2),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L4,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_eg(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9743_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or5834768355832116004an_nat(L,U)) ).

% finite_greaterThanLessThan
tff(fact_9744_isCont__Lb__Ub,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ? [L5: real,M9: real] :
            ( ! [X4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),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,L5),aa_real_real(F,X4))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),M9) ) )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),M9) )
               => ? [X2: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
                    & ( aa_real_real(F,X2) = Y2 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9745_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,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)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [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))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X4)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9746_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [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))),R3) )
               => ( aa_real_real(F,X4) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9747_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,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)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [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))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9748_isCont__inverse__function2,axiom,
    ! [A: real,Xa: real,B2: real,G3: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B2)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
               => ( aa_real_real(G3,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xa),top_top_set_real),G3) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9749_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9750_isCont__arcosh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9751_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D2: real,G3: fun_real_real,Xa: real,A: real,B2: real] :
      ( has_fi5821293074295781190e_real(F,D2,topolo2177554685111907308n_real(aa_real_real(G3,Xa),top_top_set_real))
     => ( ( D2 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B2)
           => ( ! [Y: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),B2)
                   => ( aa_real_real(F,aa_real_real(G3,Y)) = Y ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),G3)
               => has_fi5821293074295781190e_real(G3,aa_real_real(inverse_inverse_real,D2),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9752_isCont__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9753_isCont__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9754_LIM__less__bound,axiom,
    ! [B2: real,Xa: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),Xa)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(B2,Xa))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xa)) ) ) ) ).

% LIM_less_bound
tff(fact_9755_isCont__artanh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9756_isCont__inverse__function,axiom,
    ! [D: real,Xa: real,G3: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [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),Xa))),D)
           => ( aa_real_real(G3,aa_real_real(F,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),Xa))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xa),top_top_set_real),G3) ) ) ) ).

% isCont_inverse_function
tff(fact_9757_GMVT_H,axiom,
    ! [A: real,B2: real,F: fun_real_real,G3: fun_real_real,G4: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G3) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                 => has_fi5821293074295781190e_real(G3,aa_real_real(G4,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C4),B2)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A))),aa_real_real(G4,C4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G3,B2)),aa_real_real(G3,A))),aa_real_real(F4,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9758_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N3: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_eh(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9759_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N3: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_eh(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))),groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9760_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => filterlim_nat_real(aTP_Lamm_ei(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_eh(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9761_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9762_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9763_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_ej,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9764_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_9765_monoseq__convergent,axiom,
    ! [X3: fun_nat_real,B: real] :
      ( topolo6980174941875973593q_real(X3)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X3,I2))),B)
       => ~ ! [L5: real] : ~ filterlim_nat_real(X3,topolo2815343760600316023s_real(L5),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9766_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G3: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G3,aa_nat_nat(suc,N2))),aa_nat_real(G3,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G3,N2))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ek(F),G3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L6: real] :
                ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N3)),L6)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L6),at_top_nat)
                & ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_nat_real(G3,N3))
                & filterlim_nat_real(G3,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9767_LIMSEQ__inverse__zero,axiom,
    ! [X3: fun_nat_real] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X3,N2)) )
     => filterlim_nat_real(aTP_Lamm_el(X3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9768_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_em,C2),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9769_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N3)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9770_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9771_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_en(Xa),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9772_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_9773_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_9774_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_eo,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9775_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => summable_real(aTP_Lamm_eh(A)) ) ) ) ).

% summable
tff(fact_9776_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_by,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9777_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),suminf_real(aTP_Lamm_eh(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9778_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => filterlim_nat_real(aTP_Lamm_ep(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_eh(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9779_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L6: real] :
              ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))),L6)
              & filterlim_nat_real(aTP_Lamm_ep(A),topolo2815343760600316023s_real(L6),at_top_nat)
              & ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_ei(A),topolo2815343760600316023s_real(L6),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9780_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_eh(A))),groups6591440286371151544t_real(aTP_Lamm_eh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9781_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),X2)
         => ? [Y2: real] :
              ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9782_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F2: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_eq(Na),F),at_top_real,F2) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9783_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_9784_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_9785_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
         => ? [Y2: real] :
              ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9786_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F2: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_eq(Na),F),at_bot_real,F2) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9787_GMVT,axiom,
    ! [A: real,B2: real,F: fun_real_real,G3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G3) )
           => ( ! [X2: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2) )
                 => differ6690327859849518006l_real(G3,topolo2177554685111907308n_real(X2,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_fi5821293074295781190e_real(G3,G_c,topolo2177554685111907308n_real(C4,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C4,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C4),B2)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G3,B2)),aa_real_real(G3,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9788_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_cz(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9789_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_er(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9790_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_es,at_top_nat) ).

% eventually_False_sequentially
tff(fact_9791_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_nat_o(P,N) ) ) ).

% eventually_sequentially
tff(fact_9792_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X2)
         => aa_nat_o(P,X2) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9793_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( ord_le2510731241096832064er_nat(F2,at_top_nat)
    <=> ! [N7: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N7),F2) ) ).

% le_sequentially
tff(fact_9794_eventually__at__left__real,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_et(B2),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9795_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A: real,B2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(image_nat_real(F,top_top_set_nat)),set_or1222579329274155063t_real(A,B2))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_9796_Bseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xa),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9797_MVT,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ? [L6: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
              & has_fi5821293074295781190e_real(F,L6,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),L6) ) ) ) ) ) ).

% MVT
tff(fact_9798_continuous__image__closed__interval,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
       => ? [C4: real,D5: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,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_9799_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X2)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_eu(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9800_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A2,artanh_real) ) ).

% continuous_on_artanh
tff(fact_9801_Rolle__deriv,axiom,
    ! [A: real,B2: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B2) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
         => ( ! [X2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                & ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z2),X4) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9802_mvt,axiom,
    ! [A: real,B2: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi),B2)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F4,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9803_DERIV__isconst__end,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_real(F,B2) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9804_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B2)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9805_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9806_DERIV__isconst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),B2)
             => ( aa_real_real(F,Xa) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9807_Rolle,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B2) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
         => ( ! [X2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B2)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9808_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or6659071591806873216st_nat(L,U)) ).

% finite_greaterThanAtMost
tff(fact_9809_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_9810_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_9811_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_9812_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9813_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_9814_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_9815_eventually__at__right__real,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_et(A),B2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9816_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(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_9817_atLeast__0,axiom,
    set_ord_atLeast_nat(zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9818_decseq__bounded,axiom,
    ! [X3: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X3)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_nat_real(X3,I2))
       => bfun_nat_real(X3,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_9819_decseq__convergent,axiom,
    ! [X3: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X3)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_nat_real(X3,I2))
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X3,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),aa_nat_real(X3,I3)) ) ) ) ).

% decseq_convergent
tff(fact_9820_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_ord_atLeast_nat(K)),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9821_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A2,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_9822_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M5)
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M5)
         => ( gcd_Gcd_nat(M5) = lattic8265883725875713057ax_nat(comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_ev,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9823_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bv,Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_9824_bdd__above__nat,axiom,
    ! [X3: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,X3)
    <=> aa_set_nat_o(finite_finite_nat,X3) ) ).

% bdd_above_nat
tff(fact_9825_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_9826_Sup__nat__def,axiom,
    ! [X3: set_nat] :
      complete_Sup_Sup_nat(X3) = $ite(X3 = bot_bot_set_nat,zero_zero_nat,lattic8265883725875713057ax_nat(X3)) ).

% Sup_nat_def
tff(fact_9827_divide__nat__def,axiom,
    ! [M2: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Na) = $ite(Na = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ew(M2),Na)))) ).

% divide_nat_def
tff(fact_9828_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_ey,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9829_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_fa,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9830_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N7: nat] :
        ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M)
         => ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),M)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9831_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9832_card_Ocomp__fun__commute__on,axiom,
    comp_nat_nat_nat(suc,suc) = comp_nat_nat_nat(suc,suc) ).

% card.comp_fun_commute_on
tff(fact_9833_mono__times__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na)) ) ).

% mono_times_nat
tff(fact_9834_incseq__bounded,axiom,
    ! [X3: fun_nat_real,B: real] :
      ( order_mono_nat_real(X3)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X3,I2)),B)
       => bfun_nat_real(X3,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9835_incseq__convergent,axiom,
    ! [X3: fun_nat_real,B: real] :
      ( order_mono_nat_real(X3)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X3,I2)),B)
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X3,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X3,I3)),L5) ) ) ) ).

% incseq_convergent
tff(fact_9836_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9837_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ~ aa_set_nat_o(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9838_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G3)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_fc(F),G3),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9839_infinite__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N3: nat] : aa_set_nat_o(member_nat(aa_nat_nat(R3,N3)),S) ) ) ).

% infinite_enumerate
tff(fact_9840_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(F,Na)) ) ).

% strict_mono_imp_increasing
tff(fact_9841_strict__mono__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => order_5726023648592871131at_nat(infini8530281810654367211te_nat(S)) ) ).

% strict_mono_enumerate
tff(fact_9842_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_do,Na)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9843_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_9844_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F4: fun_real_real] :
      ( ! [X2: real] : has_fi5821293074295781190e_real(F,aa_real_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( ! [X2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F4,X2))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9845_inj__Suc,axiom,
    ! [N4: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N4) ).

% inj_Suc
tff(fact_9846_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( aa_set_nat_o(member_nat(N2),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dk,K)),N4) ) ).

% inj_on_diff_nat
tff(fact_9847_inj__on__set__encode,axiom,
    aa_set_set_nat_o(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9848_summable__reindex,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G3),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => summable_real(comp_nat_real_nat(F,G3)) ) ) ) ).

% summable_reindex
tff(fact_9849_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G3),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G3))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_9850_suminf__reindex,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G3),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => ( ! [X2: nat] :
                ( ~ aa_set_nat_o(member_nat(X2),image_nat_nat(G3,top_top_set_nat))
               => ( aa_nat_real(F,X2) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G3)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9851_powr__real__of__int_H,axiom,
    ! [Xa: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( ( Xa != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_int_real(ring_1_of_int_real,Na)) = aa_int_real(power_int_real(Xa),Na) ) ) ) ).

% powr_real_of_int'
tff(fact_9852_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m: nat,
                      m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                      ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_VEBT_valid(X),n2) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary3),m)
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                      & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X6)
                          & ! [X: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                             => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9853_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg4)
    <=> ( ( Deg = Deg4 )
        & $let(
            n2: nat,
            n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m: nat,
              m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
              ( ! [X: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                 => aa_nat_o(vEBT_VEBT_valid(X),n2) )
              & aa_nat_o(vEBT_VEBT_valid(Summary),m)
              & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
              & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                  & ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                     => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg),TreeList),Summary),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9854_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
      <=> (Ya) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( (Ya)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( (Ya)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(vEBT_VEBT_valid(X),n2) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary3),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9855_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m: nat,
                        m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                        ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_VEBT_valid(X),n2) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary3),m)
                        & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                        & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X6)
                            & ! [X: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                               => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9856_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Ya)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( ( (Ya)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(vEBT_VEBT_valid(X),n2) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary3),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9857_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(vEBT_VEBT_valid(X),n2) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary3),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9858_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(vEBT_VEBT_valid(X),n2) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary3),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9859_Sup__real__def,axiom,
    ! [X3: set_real] : comple1385675409528146559p_real(X3) = ord_Least_real(aTP_Lamm_fe(X3)) ).

% Sup_real_def
tff(fact_9860_Sup__int__def,axiom,
    ! [X3: set_int] : complete_Sup_Sup_int(X3) = the_int(aTP_Lamm_ff(X3)) ).

% Sup_int_def
tff(fact_9861_filterlim__lessThan__at__top,axiom,
    filter3212408913953519116et_nat(set_ord_lessThan_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).

% filterlim_lessThan_at_top
tff(fact_9862_filterlim__atMost__at__top,axiom,
    filter3212408913953519116et_nat(set_ord_atMost_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).

% filterlim_atMost_at_top
tff(fact_9863_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_fg) ).

% Rats_eq_int_div_nat
tff(fact_9864_Rats__no__top__le,axiom,
    ! [Xa: real] :
    ? [X2: real] :
      ( aa_set_real_o(member_real(X2),field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2) ) ).

% Rats_no_top_le
tff(fact_9865_Rats__no__bot__less,axiom,
    ! [Xa: real] :
    ? [X2: real] :
      ( aa_set_real_o(member_real(X2),field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Xa) ) ).

% Rats_no_bot_less
tff(fact_9866_Rats__dense__in__real,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ? [X2: real] :
          ( aa_set_real_o(member_real(X2),field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Ya) ) ) ).

% Rats_dense_in_real
tff(fact_9867_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] : bit_take_bit_num(zero_zero_nat,M2) = none_num ).

% take_bit_num_simps(1)
tff(fact_9868_take__bit__num__def,axiom,
    ! [Na: nat,M2: num] :
      bit_take_bit_num(Na,M2) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M2)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M2))))) ).

% take_bit_num_def
tff(fact_9869_num__of__nat__numeral__eq,axiom,
    ! [Q5: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q5)) = Q5 ).

% num_of_nat_numeral_eq
tff(fact_9870_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9871_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_9872_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),one_one_nat)
     => ( aa_nat_num(num_of_nat,Na) = one ) ) ).

% num_of_nat_One
tff(fact_9873_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Na)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_9874_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M2)),aa_nat_num(num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9875_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na),aa_num_num(inc,aa_nat_num(num_of_nat,Na)),one) ).

% num_of_nat.simps(2)
tff(fact_9876_num__induct,axiom,
    ! [P: fun_num_o,Xa: num] :
      ( aa_num_o(P,one)
     => ( ! [X2: num] :
            ( aa_num_o(P,X2)
           => aa_num_o(P,aa_num_num(inc,X2)) )
       => aa_num_o(P,Xa) ) ) ).

% num_induct
tff(fact_9877_add__inc,axiom,
    ! [Xa: num,Ya: num] : aa_num_num(plus_plus_num(Xa),aa_num_num(inc,Ya)) = aa_num_num(inc,aa_num_num(plus_plus_num(Xa),Ya)) ).

% add_inc
tff(fact_9878_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9879_inc_Osimps_I3_J,axiom,
    ! [Xa: num] : aa_num_num(inc,aa_num_num(bit1,Xa)) = aa_num_num(bit0,aa_num_num(inc,Xa)) ).

% inc.simps(3)
tff(fact_9880_inc_Osimps_I2_J,axiom,
    ! [Xa: num] : aa_num_num(inc,aa_num_num(bit0,Xa)) = aa_num_num(bit1,Xa) ).

% inc.simps(2)
tff(fact_9881_add__One,axiom,
    ! [Xa: num] : aa_num_num(plus_plus_num(Xa),one) = aa_num_num(inc,Xa) ).

% add_One
tff(fact_9882_inc__BitM__eq,axiom,
    ! [Na: num] : aa_num_num(inc,aa_num_num(bitM,Na)) = aa_num_num(bit0,Na) ).

% inc_BitM_eq
tff(fact_9883_BitM__inc__eq,axiom,
    ! [Na: num] : aa_num_num(bitM,aa_num_num(inc,Na)) = aa_num_num(bit1,Na) ).

% BitM_inc_eq
tff(fact_9884_mult__inc,axiom,
    ! [Xa: num,Ya: num] : aa_num_num(times_times_num(Xa),aa_num_num(inc,Ya)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(Xa),Ya)),Xa) ).

% mult_inc
tff(fact_9885_min__Suc__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),Na)) ).

% min_Suc_Suc
tff(fact_9886_min__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9887_min__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Na) = zero_zero_nat ).

% min_0L
tff(fact_9888_nat__mult__min__left,axiom,
    ! [M2: nat,Na: nat,Q5: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),Na)),Q5) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q5)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)) ).

% nat_mult_min_left
tff(fact_9889_nat__mult__min__right,axiom,
    ! [M2: nat,Na: nat,Q5: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),Q5)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q5)) ).

% nat_mult_min_right
tff(fact_9890_min__diff,axiom,
    ! [M2: nat,I: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),Na)),I) ).

% min_diff
tff(fact_9891_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9892_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_9893_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_9894_cis__Arg__unique,axiom,
    ! [Z: complex,Xa: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = cis(Xa) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => ( aa_complex_real(arg,Z) = Xa ) ) ) ) ).

% cis_Arg_unique
tff(fact_9895_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_fh(Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)),collect_complex(aTP_Lamm_br(Na))) ) ).

% bij_betw_roots_unity
tff(fact_9896_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(aa_nat_fun_real_real(root,Na),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,Na))))),collect_complex(aTP_Lamm_br(Na))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_bs(C2),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9897_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_fi(Z))) ).

% Arg_def
tff(fact_9898_bij__betw__Suc,axiom,
    ! [M5: set_nat,N4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(suc),M5),N4)
    <=> ( image_nat_nat(suc,M5) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9899_bij__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(infini8530281810654367211te_nat(S)),top_top_set_nat),S) ) ).

% bij_enumerate
tff(fact_9900_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9901_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9902_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9903_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Ya: list_int] :
      ( ( upto(Xa,Xaa) = Ya )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xa,Xaa))
       => ~ ( ( Ya = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),one_one_int),Xaa)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xa,Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9904_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9905_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9906_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9907_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9908_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9909_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M2),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na)),cons_int(aa_num_int(numeral_numeral_int,M2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),one_one_int),aa_num_int(numeral_numeral_int,Na))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9910_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,Na: num] :
      upto(uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))),cons_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),one_one_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9911_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,Na: num] :
      upto(uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,Na)),cons_int(uminus_uminus_int(aa_num_int(numeral_numeral_int,M2)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,uminus_uminus_int(aa_num_int(numeral_numeral_int,M2))),one_one_int),aa_num_int(numeral_numeral_int,Na))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9912_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M2),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na))),cons_int(aa_num_int(numeral_numeral_int,M2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),one_one_int),uminus_uminus_int(aa_num_int(numeral_numeral_int,Na)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9913_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9914_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9915_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9916_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Ya: list_int] :
      ( ( upto(Xa,Xaa) = Ya )
     => ( Ya = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9917_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9918_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9919_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9920_card__length__sum__list__rec,axiom,
    ! [M2: nat,N4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M2)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_fj(M2),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_fk(M2),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_fl(M2),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9921_less__eq__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_fn),rep_Integ(Xa)),rep_Integ(Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9922_less__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_fp),rep_Integ(Xa)),rep_Integ(Xaa)) ) ).

% less_int.rep_eq
tff(fact_9923_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb3: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,abs_Integ(Xa)),abs_Integ(Xb3))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_fn),Xa),Xb3) ) ).

% less_eq_int.abs_eq
tff(fact_9924_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb3: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,abs_Integ(Xa)),abs_Integ(Xb3))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_fp),Xa),Xb3) ) ).

% less_int.abs_eq
tff(fact_9925_zero__int__def,axiom,
    zero_zero_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9926_int__def,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,Na) = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Na),zero_zero_nat)) ).

% int_def
tff(fact_9927_one__int__def,axiom,
    one_one_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9928_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9929_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9930_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_9931_take__upt,axiom,
    ! [I: nat,M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)),Na)
     => ( take_nat(M2,upt(I,Na)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)) ) ) ).

% take_upt
tff(fact_9932_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9933_upt__rec__numeral,axiom,
    ! [M2: num,Na: num] :
      upt(aa_num_nat(numeral_numeral_nat,M2),aa_num_nat(numeral_numeral_nat,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na)),cons_nat(aa_num_nat(numeral_numeral_nat,M2),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Na))),nil_nat) ).

% upt_rec_numeral
tff(fact_9934_sum__list__upt,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( groups4561878855575611511st_nat(upt(M2,Na)) = groups3542108847815614940at_nat(aTP_Lamm_cd,set_or4665077453230672383an_nat(M2,Na)) ) ) ).

% sum_list_upt
tff(fact_9935_map__add__upt,axiom,
    ! [Na: nat,M2: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fq,Na),upt(zero_zero_nat,M2)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Na)) ).

% map_add_upt
tff(fact_9936_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9937_sorted__wrt__upt,axiom,
    ! [M2: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(M2,Na)) ).

% sorted_wrt_upt
tff(fact_9938_sorted__upt,axiom,
    ! [M2: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(M2,Na)) ).

% sorted_upt
tff(fact_9939_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9940_atLeast__upt,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Na) = set_nat2(upt(zero_zero_nat,Na)) ).

% atLeast_upt
tff(fact_9941_atMost__upto,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,Na) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Na))) ).

% atMost_upto
tff(fact_9942_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9943_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_9944_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9945_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9946_map__decr__upt,axiom,
    ! [M2: nat,Na: nat] : map_nat_nat(aTP_Lamm_fr,upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,Na))) = upt(M2,Na) ).

% map_decr_upt
tff(fact_9947_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xa,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xa )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9948_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_9949_sorted__upto,axiom,
    ! [M2: int,Na: int] : sorted_wrt_int(ord_less_eq_int,upto(M2,Na)) ).

% sorted_upto
tff(fact_9950_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt_int(ord_less_int,upto(I,J)) ).

% sorted_wrt_upto
tff(fact_9951_Field__natLeq__on,axiom,
    ! [Na: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Na)) ).

% Field_natLeq_on
tff(fact_9952_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_9953_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_9954_cauchy__def,axiom,
    ! [X3: fun_nat_rat] :
      ( cauchy(X3)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X3,M)),aa_nat_rat(X3,N)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_9955_cauchyI,axiom,
    ! [X3: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K8: nat] :
            ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K8),M4)
             => ! [N2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K8),N2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X3,M4)),aa_nat_rat(X3,N2)))),R3) ) ) )
     => cauchy(X3) ) ).

% cauchyI
tff(fact_9956_cauchy__imp__bounded,axiom,
    ! [X3: fun_nat_rat] :
      ( cauchy(X3)
     => ? [B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
          & ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X3,N3))),B4) ) ) ).

% cauchy_imp_bounded
tff(fact_9957_cauchyD,axiom,
    ! [X3: fun_nat_rat,R2: rat] :
      ( cauchy(X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K2: nat] :
          ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M3)
           => ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X3,M3)),aa_nat_rat(X3,N3)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_9958_le__Real,axiom,
    ! [X3: fun_nat_rat,Y3: fun_nat_rat] :
      ( cauchy(X3)
     => ( cauchy(Y3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real2(X3)),real2(Y3))
        <=> ! [R5: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
             => ? [K3: nat] :
                ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X3,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y3,N)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_9959_cauchy__not__vanishes,axiom,
    ! [X3: fun_nat_rat] :
      ( cauchy(X3)
     => ( ~ vanishes(X3)
       => ? [B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
            & ? [K2: nat] :
              ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_rat_rat(abs_abs_rat,aa_nat_rat(X3,N3))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9960_vanishes__mult__bounded,axiom,
    ! [X3: fun_nat_rat,Y3: fun_nat_rat] :
      ( ? [A8: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A8)
          & ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X3,N2))),A8) )
     => ( vanishes(Y3)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ft(X3),Y3)) ) ) ).

% vanishes_mult_bounded
tff(fact_9961_vanishesD,axiom,
    ! [X3: fun_nat_rat,R2: rat] :
      ( vanishes(X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K2: nat] :
          ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X3,N3))),R2) ) ) ) ).

% vanishesD
tff(fact_9962_vanishesI,axiom,
    ! [X3: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K8: nat] :
            ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K8),N2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X3,N2))),R3) ) )
     => vanishes(X3) ) ).

% vanishesI
tff(fact_9963_vanishes__def,axiom,
    ! [X3: fun_nat_rat] :
      ( vanishes(X3)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X3,N))),R5) ) ) ) ).

% vanishes_def
tff(fact_9964_cauchy__not__vanishes__cases,axiom,
    ! [X3: fun_nat_rat] :
      ( cauchy(X3)
     => ( ~ vanishes(X3)
       => ? [B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
            & ? [K2: nat] :
                ( ! [N3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X3,N3))) )
                | ! [N3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_nat_rat(X3,N3)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9965_not__positive__Real,axiom,
    ! [X3: fun_nat_rat] :
      ( cauchy(X3)
     => ( ~ aa_real_o(positive2,real2(X3))
      <=> ! [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
           => ? [K3: nat] :
              ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X3,N)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_9966_positive__Real,axiom,
    ! [X3: fun_nat_rat] :
      ( cauchy(X3)
     => ( aa_real_o(positive2,real2(X3))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(X3,N)) ) ) ) ) ).

% positive_Real
tff(fact_9967_less__real__def,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
    <=> aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Xa)) ) ).

% less_real_def
tff(fact_9968_Real_Opositive_Orep__eq,axiom,
    ! [Xa: real] :
      ( aa_real_o(positive2,Xa)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,Xa),N)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9969_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_9970_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_9971_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% mask_nonnegative_int
tff(fact_9972_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9973_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_9974_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% mask_nat_less_exp
tff(fact_9975_rat__less__eq__code,axiom,
    ! [P3: rat,Q5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),Q5)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_fv(Q5)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_9976_of__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( real_V4546457046886955230omplex(aa_real_real(sqrt,Xa)) = aa_complex_complex(csqrt,real_V4546457046886955230omplex(Xa)) ) ) ).

% of_real_sqrt
tff(fact_9977_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q5: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q5) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q5) ) ).

% quotient_of_denom_pos
tff(fact_9978_rat__less__code,axiom,
    ! [P3: rat,Q5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),Q5)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_fx(Q5)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_9979_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K: num] :
      set_or4665077453230672383an_nat(M2,aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(M2,aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9980_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_9981_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa_nat_nat(suc,Na) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Na = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_9982_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Na) )
    <=> ( aa_num_nat(pred_numeral,K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_9983_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_9984_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% less_numeral_Suc
tff(fact_9985_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_9986_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_9987_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% le_numeral_Suc
tff(fact_9988_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_9989_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Na) ).

% diff_numeral_Suc
tff(fact_9990_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_9991_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9992_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% max_numeral_Suc
tff(fact_9993_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_9994_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% min_numeral_Suc
tff(fact_9995_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9996_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_9997_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_9998_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_9999_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_10000_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_fy),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_10001_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_es),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_10002_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),Na)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M2)),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_10003_max__Suc2,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_fz,Na)),M2) ).

% max_Suc2
tff(fact_10004_max__Suc1,axiom,
    ! [Na: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),M2) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_ga,Na)),M2) ).

% max_Suc1
tff(fact_10005_diff__Suc,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cd),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)) ).

% diff_Suc
tff(fact_10006_min__Suc2,axiom,
    ! [M2: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_gb,Na)),M2) ).

% min_Suc2
tff(fact_10007_min__Suc1,axiom,
    ! [Na: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),M2) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_gc,Na)),M2) ).

% min_Suc1
tff(fact_10008_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cd),Nat) ).

% pred_def
tff(fact_10009_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_10010_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri7919022796975470100ot_int(K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_10011_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_ri7919022796975470100ot_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_10012_Real_Opositive_Oabs__eq,axiom,
    ! [Xa: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xa),Xa)
     => ( aa_real_o(positive2,real2(Xa))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Xa,N)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_10013_sqr_Osimps_I3_J,axiom,
    ! [Na: num] : aa_num_num(sqr,aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(sqr,Na)),Na))) ).

% sqr.simps(3)
tff(fact_10014_sqr__conv__mult,axiom,
    ! [Xa: num] : aa_num_num(sqr,Xa) = aa_num_num(times_times_num(Xa),Xa) ).

% sqr_conv_mult
tff(fact_10015_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_10016_sqr_Osimps_I2_J,axiom,
    ! [Na: num] : aa_num_num(sqr,aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Na))) ).

% sqr.simps(2)
tff(fact_10017_pow_Osimps_I3_J,axiom,
    ! [Xa: num,Ya: num] : aa_num_num(pow(Xa),aa_num_num(bit1,Ya)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(Xa),Ya))),Xa) ).

% pow.simps(3)
tff(fact_10018_Real_Opositive_Orsp,axiom,
    bNF_re7504242029904815401at_o_o(realrel,fequal_o,aTP_Lamm_gd,aTP_Lamm_gd) ).

% Real.positive.rsp
tff(fact_10019_pow_Osimps_I1_J,axiom,
    ! [Xa: num] : aa_num_num(pow(Xa),one) = Xa ).

% pow.simps(1)
tff(fact_10020_pow_Osimps_I2_J,axiom,
    ! [Xa: num,Ya: num] : aa_num_num(pow(Xa),aa_num_num(bit0,Ya)) = aa_num_num(sqr,aa_num_num(pow(Xa),Ya)) ).

% pow.simps(2)
tff(fact_10021_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_10022_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_10023_less__eq__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_10024_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_10025_Real_Opositive_Otransfer,axiom,
    bNF_re3218747500659316912al_o_o(pcr_real,fequal_o,aTP_Lamm_gd,positive2) ).

% Real.positive.transfer
tff(fact_10026_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_fn),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_10027_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_fp),ord_less_int) ).

% less_int.transfer
tff(fact_10028_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_10029_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_ge,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_10030_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_10031_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,Na: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),Na),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% pred_nat_trancl_eq_le
tff(fact_10032_less__eq,axiom,
    ! [M2: nat,Na: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),Na),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% less_eq
tff(fact_10033_finite__vimage__Suc__iff,axiom,
    ! [F2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F2))
    <=> aa_set_nat_o(finite_finite_nat,F2) ) ).

% finite_vimage_Suc_iff
tff(fact_10034_vimage__Suc__insert__Suc,axiom,
    ! [Na: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Na),A2)) = insert_nat(Na,vimage_nat_nat(suc,A2)) ).

% vimage_Suc_insert_Suc
tff(fact_10035_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_10036_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_gf(M2))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M2),aTP_Lamm_gg(M2)) ).

% pairs_le_eq_Sigma
tff(fact_10037_Restr__natLeq,axiom,
    ! [Na: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Na)),aTP_Lamm_gh(Na))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na))) ).

% Restr_natLeq
tff(fact_10038_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_10039_natLeq__on__wo__rel,axiom,
    ! [Na: nat] : bNF_We3818239936649020644el_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na)))) ).

% natLeq_on_wo_rel
tff(fact_10040_Restr__natLeq2,axiom,
    ! [Na: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,Na),aTP_Lamm_gi(Na))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na))) ).

% Restr_natLeq2
tff(fact_10041_natLeq__underS__less,axiom,
    ! [Na: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,Na) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Na)) ).

% natLeq_underS_less
tff(fact_10042_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_10043_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_10044_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_10045_bezw__non__0,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ya)
     => ( bezw(Xa,Ya) = product_Pair_int_int(product_snd_int_int(bezw(Ya,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Ya,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Ya,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Ya))))) ) ) ).

% bezw_non_0
tff(fact_10046_bezw_Osimps,axiom,
    ! [Xa: nat,Ya: nat] :
      bezw(Xa,Ya) = $ite(Ya = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Ya,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Ya,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Ya,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Ya)))))) ).

% bezw.simps
tff(fact_10047_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod_int_int] :
      ( ( bezw(Xa,Xaa) = Ya )
     => ( Ya = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_10048_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod_int_int] :
      ( ( bezw(Xa,Xaa) = Ya )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Ya = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_10049_Rat_Opositive_Orep__eq,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(positive,Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xa))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xa)))) ) ).

% Rat.positive.rep_eq
tff(fact_10050_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      normalize(P3) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P3)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)),
          product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a)) ),
        $ite(
          product_snd_int_int(P3) = zero_zero_int,
          product_Pair_int_int(zero_zero_int,one_one_int),
          $let(
            a: int,
            a:= uminus_uminus_int(aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))),
            product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a)) ) ) ) ).

% normalize_def
tff(fact_10051_gcd__pos__int,axiom,
    ! [M2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M2),Na))
    <=> ( ( M2 != zero_zero_int )
        | ( Na != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_10052_gcd__ge__0__int,axiom,
    ! [Xa: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Ya)) ).

% gcd_ge_0_int
tff(fact_10053_gcd__le1__int,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B2)),A) ) ).

% gcd_le1_int
tff(fact_10054_gcd__le2__int,axiom,
    ! [B2: int,A: 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(gcd_gcd_int,A),B2)),B2) ) ).

% gcd_le2_int
tff(fact_10055_gcd__cases__int,axiom,
    ! [Xa: int,Ya: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Ya)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),uminus_uminus_int(Ya))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,uminus_uminus_int(Xa)),Ya)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,uminus_uminus_int(Xa)),uminus_uminus_int(Ya))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Ya)) ) ) ) ) ).

% gcd_cases_int
tff(fact_10056_gcd__unique__int,axiom,
    ! [D: int,A: int,B2: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B2)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B2) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B2) ) ) ).

% gcd_unique_int
tff(fact_10057_gcd__non__0__int,axiom,
    ! [Ya: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ya)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa),Ya) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Ya),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Ya)) ) ) ).

% gcd_non_0_int
tff(fact_10058_Rat_Opositive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_gj,positive) ).

% Rat.positive.transfer
tff(fact_10059_quotient__of__def,axiom,
    ! [Xa: rat] : quotient_of(Xa) = the_Pr4378521158711661632nt_int(aTP_Lamm_gk(Xa)) ).

% quotient_of_def
tff(fact_10060_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_10061_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_10062_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2) )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_10063_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_10064_gcd__0__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),zero_zero_nat) = Xa ).

% gcd_0_nat
tff(fact_10065_gcd__0__left__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xa) = Xa ).

% gcd_0_left_nat
tff(fact_10066_gcd__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_10067_gcd__pos__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),Na))
    <=> ( ( M2 != zero_zero_nat )
        | ( Na != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_10068_normalize__stable,axiom,
    ! [Q5: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q5)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P3),Q5)
       => ( normalize(product_Pair_int_int(P3,Q5)) = product_Pair_int_int(P3,Q5) ) ) ) ).

% normalize_stable
tff(fact_10069_gcd__diff2__nat,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),Na) ) ) ).

% gcd_diff2_nat
tff(fact_10070_gcd__diff1__nat,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),Na)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),Na) ) ) ).

% gcd_diff1_nat
tff(fact_10071_gcd__le1__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2)),A) ) ).

% gcd_le1_nat
tff(fact_10072_gcd__le2__nat,axiom,
    ! [B2: nat,A: 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(gcd_gcd_nat,A),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_10073_gcd__non__0__nat,axiom,
    ! [Ya: nat,Xa: nat] :
      ( ( Ya != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya)) ) ) ).

% gcd_non_0_nat
tff(fact_10074_gcd__nat_Osimps,axiom,
    ! [Xa: nat,Ya: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Ya) = $ite(Ya = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ya))) ).

% gcd_nat.simps
tff(fact_10075_gcd__nat_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Ya )
     => ( Ya = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_10076_bezout__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [X2: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2)) ) ).

% bezout_nat
tff(fact_10077_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A3: nat,B4: nat] :
          ( aa_set_nat_o(member_nat(A3),A2)
         => ( aa_set_nat_o(member_nat(B4),A2)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B4)),A2) ) )
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(gcd_Gcd_nat(A2)),A2) ) ) ).

% Gcd_in
tff(fact_10078_Rat__cases,axiom,
    ! [Q5: rat] :
      ~ ! [A3: int,B4: int] :
          ( ( Q5 = aa_int_rat(fract(A3),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B4) ) ) ).

% Rat_cases
tff(fact_10079_Rat__induct,axiom,
    ! [P: fun_rat_o,Q5: rat] :
      ( ! [A3: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B4)
           => aa_rat_o(P,aa_int_rat(fract(A3),B4)) ) )
     => aa_rat_o(P,Q5) ) ).

% Rat_induct
tff(fact_10080_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A: nat] :
    ? [X2: nat,Y: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_10081_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : gcd_Gcd_nat(set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_10082_Rat__cases__nonzero,axiom,
    ! [Q5: rat] :
      ( ! [A3: int,B4: int] :
          ( ( Q5 = aa_int_rat(fract(A3),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ( ( A3 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B4) ) ) )
     => ( Q5 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_10083_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_gl) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_10084_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),Na) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gm(Na),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_10085_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X2: product_prod_int_int] :
      ( ( R2 = aa_int_rat(fract(product_fst_int_int(X2)),product_snd_int_int(X2)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(X2))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(X2)),product_snd_int_int(X2))
      & ! [Y2: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(fract(product_fst_int_int(Y2)),product_snd_int_int(Y2)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Y2))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Y2)),product_snd_int_int(Y2)) )
         => ( Y2 = X2 ) ) ) ).

% quotient_of_unique
tff(fact_10086_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Ya )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Ya = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_10087_coprime__Suc__0__left,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ).

% coprime_Suc_0_left
tff(fact_10088_coprime__Suc__0__right,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_10089_coprime__diff__one__right__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_10090_coprime__diff__one__left__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),Na) ) ).

% coprime_diff_one_left_nat
tff(fact_10091_Rats__abs__nat__div__natE,axiom,
    ! [Xa: real] :
      ( aa_set_real_o(member_real(Xa),field_5140801741446780682s_real)
     => ~ ! [M4: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M4)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M4),N2) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_10092_Rat_Opositive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_gj,aTP_Lamm_gj) ).

% Rat.positive.rsp
tff(fact_10093_Rat_Opositive_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_rat_o(positive,abs_Rat(Xa))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa)),product_snd_int_int(Xa))) ) ) ).

% Rat.positive.abs_eq
tff(fact_10094_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gn(F),M5),at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),M4)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,M4)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_10095_Bseq__mono__convergent,axiom,
    ! [X3: fun_nat_real] :
      ( bfun_nat_real(X3,at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X3,M4)),aa_nat_real(X3,N2)) )
       => topolo7531315842566124627t_real(X3) ) ) ).

% Bseq_mono_convergent
tff(fact_10096_convergent__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => topolo7531315842566124627t_real(aa_real_fun_nat_real(power_power_real,Xa)) ) ) ).

% convergent_realpow
tff(fact_10097_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gn(F),M5),at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),M4)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,M4)),aa_nat_real(F,N2)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_10098_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_10099_pair__less__iff1,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Ya),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Z) ) ).

% pair_less_iff1
tff(fact_10100_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_10101_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S3),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_10102_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_10103_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_gl),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_10104_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_bh,aTP_Lamm_bg),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_10105_bit__concat__bit__iff,axiom,
    ! [M2: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_concat_bit(M2),K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M2)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_10106_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_10107_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_10108_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Na),K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_10109_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_fn),produc6395135316500328296_nat_o(aTP_Lamm_fn)) ).

% less_eq_int.rsp
tff(fact_10110_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_fp),produc6395135316500328296_nat_o(aTP_Lamm_fp)) ).

% less_int.rsp
tff(fact_10111_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int.rsp
tff(fact_10112_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int.rsp
tff(fact_10113_less__eq__enat__def,axiom,
    ! [M2: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,M2),Na)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_go(M2),$true),Na) ) ).

% less_eq_enat_def
tff(fact_10114_less__enat__def,axiom,
    ! [M2: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,M2),Na)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_gp(Na),$false),M2) ) ).

% less_enat_def
tff(fact_10115_division__segment__int__def,axiom,
    ! [K: int] :
      euclid3395696857347342551nt_int(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K),one_one_int,uminus_uminus_int(one_one_int)) ).

% division_segment_int_def
tff(fact_10116_of__rat__dense,axiom,
    ! [Xa: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Ya)
     => ? [Q3: rat] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_rat_real(field_7254667332652039916t_real,Q3))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q3)),Ya) ) ) ).

% of_rat_dense
tff(fact_10117_less__RealD,axiom,
    ! [Y3: fun_nat_rat,Xa: real] :
      ( cauchy(Y3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),real2(Y3))
       => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y3,N2))) ) ) ).

% less_RealD
tff(fact_10118_le__RealI,axiom,
    ! [Y3: fun_nat_rat,Xa: real] :
      ( cauchy(Y3)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y3,N2)))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),real2(Y3)) ) ) ).

% le_RealI
tff(fact_10119_Real__leI,axiom,
    ! [X3: fun_nat_rat,Ya: real] :
      ( cauchy(X3)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X3,N2))),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real2(X3)),Ya) ) ) ).

% Real_leI
tff(fact_10120_compute__powr__real,axiom,
    ! [B2: real,I: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B2),I) = $ite(
        aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real),
        abort_real(literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$false,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_gq(B2),I)),
        $ite(
          aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I,
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I)))))),
          abort_real(literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$true,$false,$true,$false,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$true,$true,$false,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$true,$true,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_gq(B2),I)) ) ) ).

% compute_powr_real
tff(fact_10121_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_10122_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_10123_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_10124_nat__of__char__less__256,axiom,
    ! [C2: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_10125_less__than__iff,axiom,
    ! [Xa: nat,Ya: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Ya),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ya) ) ).

% less_than_iff
tff(fact_10126_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),Na)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimnum
tff(fact_10127_enat__ord__simps_I2_J,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extended_enat2(M2)),extended_enat2(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na) ) ).

% enat_ord_simps(2)
tff(fact_10128_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,Na: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extended_enat2(M2)),extended_enat2(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),Na) ) ).

% enat_ord_simps(1)
tff(fact_10129_idiff__enat__0__right,axiom,
    ! [Na: extended_enat] : minus_3235023915231533773d_enat(Na,extended_enat2(zero_zero_nat)) = Na ).

% idiff_enat_0_right
tff(fact_10130_idiff__enat__0,axiom,
    ! [Na: extended_enat] : minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat),Na) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_10131_numeral__less__enat__iff,axiom,
    ! [M2: num,Na: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M2)),extended_enat2(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),Na) ) ).

% numeral_less_enat_iff
tff(fact_10132_numeral__le__enat__iff,axiom,
    ! [M2: num,Na: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M2)),extended_enat2(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),Na) ) ).

% numeral_le_enat_iff
tff(fact_10133_finite__enat__bounded,axiom,
    ! [A2: set_Extended_enat,Na: nat] :
      ( ! [Y: extended_enat] :
          ( aa_set3337841544853333262enat_o(member_Extended_enat(Y),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),extended_enat2(Na)) )
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ).

% finite_enat_bounded
tff(fact_10134_enat__ile,axiom,
    ! [Na: extended_enat,M2: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Na),extended_enat2(M2))
     => ? [K2: nat] : Na = extended_enat2(K2) ) ).

% enat_ile
tff(fact_10135_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Uu: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf((A),(B2)),Uu) = vEBT_Leaf((A),(B2)) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_10136_enat__iless,axiom,
    ! [Na: extended_enat,M2: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Na),extended_enat2(M2))
     => ? [K2: nat] : Na = extended_enat2(K2) ) ).

% enat_iless
tff(fact_10137_less__enatE,axiom,
    ! [Na: extended_enat,M2: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Na),extended_enat2(M2))
     => ~ ! [K2: nat] :
            ( ( Na = extended_enat2(K2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),M2) ) ) ).

% less_enatE
tff(fact_10138_Suc__ile__eq,axiom,
    ! [M2: nat,Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extended_enat2(aa_nat_nat(suc,M2))),Na)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extended_enat2(M2)),Na) ) ).

% Suc_ile_eq
tff(fact_10139_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_10140_enat__0__iff_I1_J,axiom,
    ! [Xa: nat] :
      ( ( extended_enat2(Xa) = zero_z5237406670263579293d_enat )
    <=> ( Xa = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_10141_enat__0__iff_I2_J,axiom,
    ! [Xa: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(Xa) )
    <=> ( Xa = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_10142_iadd__le__enat__iff,axiom,
    ! [Xa: extended_enat,Ya: extended_enat,Na: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,plus_p3455044024723400733d_enat(Xa,Ya)),extended_enat2(Na))
    <=> ? [Y5: nat,X7: nat] :
          ( ( Xa = extended_enat2(X7) )
          & ( Ya = extended_enat2(Y5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X7),Y5)),Na) ) ) ).

% iadd_le_enat_iff
tff(fact_10143_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: extended_enat,Ya: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xa,Xaa) = Ya )
     => ( ! [A3: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B4)) )
           => ( Ya != vEBT_Leaf((A3),(B4)) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
             => ( ( Xaa = extend5688581933313929465d_enat )
               => ( Ya != vEBT_Node(Info2,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_gr(Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary3,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
               => ! [L6: nat] :
                    ( ( Xaa = extended_enat2(L6) )
                   => ( Ya != vEBT_Node(Info2,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_gr(Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary3,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_10144_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),Na)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimcomplete
tff(fact_10145_enat__ord__simps_I6_J,axiom,
    ! [Q5: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extend5688581933313929465d_enat),Q5) ).

% enat_ord_simps(6)
tff(fact_10146_enat__ord__simps_I4_J,axiom,
    ! [Q5: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Q5),extend5688581933313929465d_enat)
    <=> ( Q5 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
tff(fact_10147_enat__ord__code_I3_J,axiom,
    ! [Q5: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Q5),extend5688581933313929465d_enat) ).

% enat_ord_code(3)
tff(fact_10148_enat__ord__simps_I5_J,axiom,
    ! [Q5: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extend5688581933313929465d_enat),Q5)
    <=> ( Q5 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
tff(fact_10149_times__enat__simps_I4_J,axiom,
    ! [M2: nat] :
      times_7803423173614009249d_enat(extended_enat2(M2),extend5688581933313929465d_enat) = $ite(M2 = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(4)
tff(fact_10150_times__enat__simps_I3_J,axiom,
    ! [Na: nat] :
      times_7803423173614009249d_enat(extend5688581933313929465d_enat,extended_enat2(Na)) = $ite(Na = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(3)
tff(fact_10151_enat__ord__code_I5_J,axiom,
    ! [Na: nat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extend5688581933313929465d_enat),extended_enat2(Na)) ).

% enat_ord_code(5)
tff(fact_10152_infinity__ileE,axiom,
    ! [M2: nat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extend5688581933313929465d_enat),extended_enat2(M2)) ).

% infinity_ileE
tff(fact_10153_enat__ord__code_I4_J,axiom,
    ! [M2: nat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extended_enat2(M2)),extend5688581933313929465d_enat) ).

% enat_ord_code(4)
tff(fact_10154_less__infinityE,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Na),extend5688581933313929465d_enat)
     => ~ ! [K2: nat] : Na != extended_enat2(K2) ) ).

% less_infinityE
tff(fact_10155_infinity__ilessE,axiom,
    ! [M2: nat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extend5688581933313929465d_enat),extended_enat2(M2)) ).

% infinity_ilessE
tff(fact_10156_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [Xa: produc7272778201969148633d_enat] :
      ( ! [A3: $o,B4: $o,Uu2: extended_enat] : Xa != produc581526299967858633d_enat(vEBT_Leaf((A3),(B4)),Uu2)
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] : Xa != produc581526299967858633d_enat(vEBT_Node(Info2,Deg2,TreeList2,Summary3),extend5688581933313929465d_enat)
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,L6: nat] : Xa != produc581526299967858633d_enat(vEBT_Node(Info2,Deg2,TreeList2,Summary3),extended_enat2(L6)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_10157_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B2: extended_enat,C2: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,plus_p3455044024723400733d_enat(A,B2)),plus_p3455044024723400733d_enat(A,C2))
    <=> ( ( A != extend5688581933313929465d_enat )
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B2),C2) ) ) ).

% enat_add_left_cancel_less
tff(fact_10158_enat__ord__simps_I3_J,axiom,
    ! [Q5: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Q5),extend5688581933313929465d_enat) ).

% enat_ord_simps(3)
tff(fact_10159_enat__add__left__cancel__le,axiom,
    ! [A: extended_enat,B2: extended_enat,C2: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,plus_p3455044024723400733d_enat(A,B2)),plus_p3455044024723400733d_enat(A,C2))
    <=> ( ( A = extend5688581933313929465d_enat )
        | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B2),C2) ) ) ).

% enat_add_left_cancel_le
tff(fact_10160_imult__infinity__right,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Na)
     => ( times_7803423173614009249d_enat(Na,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_10161_imult__infinity,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Na)
     => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,Na) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_10162_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: extended_enat,Ya: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xa,Xaa) = Ya )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(Xa,Xaa))
       => ( ! [A3: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B4)) )
             => ( ( Ya = vEBT_Leaf((A3),(B4)) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Leaf((A3),(B4)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
               => ( ( Xaa = extend5688581933313929465d_enat )
                 => ( ( Ya = vEBT_Node(Info2,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_gr(Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary3,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info2,Deg2,TreeList2,Summary3),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
                 => ! [L6: nat] :
                      ( ( Xaa = extended_enat2(L6) )
                     => ( ( Ya = vEBT_Node(Info2,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_gr(Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary3,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info2,Deg2,TreeList2,Summary3),extended_enat2(L6))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_10163_times__enat__def,axiom,
    ! [M2: extended_enat,Na: extended_enat] :
      times_7803423173614009249d_enat(M2,Na) = extend3600170679010898289d_enat(aTP_Lamm_gt(Na),
        $ite(Na = zero_z5237406670263579293d_enat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),
        M2) ).

% times_enat_def
tff(fact_10164_iless__Suc__eq,axiom,
    ! [M2: nat,Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extended_enat2(M2)),extended_eSuc(Na))
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extended_enat2(M2)),Na) ) ).

% iless_Suc_eq
tff(fact_10165_eSuc__mono,axiom,
    ! [Na: extended_enat,M2: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,extended_eSuc(Na)),extended_eSuc(M2))
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Na),M2) ) ).

% eSuc_mono
tff(fact_10166_eSuc__ile__mono,axiom,
    ! [Na: extended_enat,M2: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extended_eSuc(Na)),extended_eSuc(M2))
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Na),M2) ) ).

% eSuc_ile_mono
tff(fact_10167_iless__eSuc0,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Na),extended_eSuc(zero_z5237406670263579293d_enat))
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
tff(fact_10168_ile__eSuc,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Na),extended_eSuc(Na)) ).

% ile_eSuc
tff(fact_10169_not__eSuc__ilei0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extended_eSuc(Na)),zero_z5237406670263579293d_enat) ).

% not_eSuc_ilei0
tff(fact_10170_i0__iless__eSuc,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),extended_eSuc(Na)) ).

% i0_iless_eSuc
tff(fact_10171_ileI1,axiom,
    ! [M2: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,M2),Na)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,extended_eSuc(M2)),Na) ) ).

% ileI1
tff(fact_10172_natLeq__on__well__order__on,axiom,
    ! [Na: nat] : order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Na)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na)))) ).

% natLeq_on_well_order_on
tff(fact_10173_natLeq__on__Well__order,axiom,
    ! [Na: nat] : order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na)))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_fs(Na)))) ).

% natLeq_on_Well_order
tff(fact_10174_Real_Opositive__def,axiom,
    positive2 = map_fu6719675619867407465at_o_o(rep_real,id_o,aTP_Lamm_gd) ).

% Real.positive_def
tff(fact_10175_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_10176_complex__Re__le__cmod,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xa)),aa_complex_real(real_V1022390504157884413omplex,Xa)) ).

% complex_Re_le_cmod
tff(fact_10177_abs__Re__le__cmod,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xa))),aa_complex_real(real_V1022390504157884413omplex,Xa)) ).

% abs_Re_le_cmod
tff(fact_10178_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_10179_Rat_Opositive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_gj) ).

% Rat.positive_def
tff(fact_10180_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_10181_csqrt__unique,axiom,
    ! [W2: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),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,W2))
          | ( ( aa_complex_real(re,W2) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W2)) ) )
       => ( aa_complex_complex(csqrt,Z) = W2 ) ) ) ).

% csqrt_unique
tff(fact_10182_csqrt__of__real__nonneg,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_real(im,Xa) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xa))
       => ( aa_complex_complex(csqrt,Xa) = real_V4546457046886955230omplex(aa_real_real(sqrt,aa_complex_real(re,Xa))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_10183_abs__Im__le__cmod,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xa))),aa_complex_real(real_V1022390504157884413omplex,Xa)) ).

% abs_Im_le_cmod
tff(fact_10184_cmod__Im__le__iff,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_real(re,Xa) = aa_complex_real(re,Ya) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),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,Xa))),aa_real_real(abs_abs_real,aa_complex_real(im,Ya))) ) ) ).

% cmod_Im_le_iff
tff(fact_10185_cmod__Re__le__iff,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( aa_complex_real(im,Xa) = aa_complex_real(im,Ya) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xa)),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,Xa))),aa_real_real(abs_abs_real,aa_complex_real(re,Ya))) ) ) ).

% cmod_Re_le_iff
tff(fact_10186_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_10187_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_10188_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_10189_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_10190_csqrt__of__real__nonpos,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_real(im,Xa) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xa)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xa) = aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xa))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_10191_csqrt__minus,axiom,
    ! [Xa: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xa)),zero_zero_real)
        | ( ( aa_complex_real(im,Xa) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xa)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xa)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xa)) ) ) ).

% csqrt_minus
tff(fact_10192_complex__div__gt__0,axiom,
    ! [A: 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(A),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(A),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(A),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(A),aa_complex_complex(cnj,B2)))) ) ) ).

% complex_div_gt_0
tff(fact_10193_Re__complex__div__gt__0,axiom,
    ! [A: 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(A),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(A),aa_complex_complex(cnj,B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_10194_Re__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),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(A),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_10195_Re__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),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(A),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_10196_Re__complex__div__ge__0,axiom,
    ! [A: 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(A),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(A),aa_complex_complex(cnj,B2)))) ) ).

% Re_complex_div_ge_0
tff(fact_10197_Im__complex__div__gt__0,axiom,
    ! [A: 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(A),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(A),aa_complex_complex(cnj,B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_10198_Im__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),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(A),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_10199_Im__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),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(A),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_10200_Im__complex__div__ge__0,axiom,
    ! [A: 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(A),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(A),aa_complex_complex(cnj,B2)))) ) ).

% Im_complex_div_ge_0
tff(fact_10201_MOST__ge__nat,axiom,
    ! [M2: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,M2),cofinite_nat) ).

% MOST_ge_nat
tff(fact_10202_MOST__nat__le,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(P,N) ) ) ).

% MOST_nat_le
tff(fact_10203_MOST__Suc__iff,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_cz(P),cofinite_nat)
    <=> eventually_nat(P,cofinite_nat) ) ).

% MOST_Suc_iff
tff(fact_10204_MOST__SucI,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,cofinite_nat)
     => eventually_nat(aTP_Lamm_cz(P),cofinite_nat) ) ).

% MOST_SucI
tff(fact_10205_MOST__SucD,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_cz(P),cofinite_nat)
     => eventually_nat(P,cofinite_nat) ) ).

% MOST_SucD
tff(fact_10206_cofinite__eq__sequentially,axiom,
    cofinite_nat = at_top_nat ).

% cofinite_eq_sequentially
tff(fact_10207_MOST__nat,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
         => aa_nat_o(P,N) ) ) ).

% MOST_nat
tff(fact_10208_INFM__nat,axiom,
    ! [P: fun_nat_o] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
          & aa_nat_o(P,N) ) ) ).

% INFM_nat
tff(fact_10209_INFM__nat__le,axiom,
    ! [P: fun_nat_o] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_nat_o(P,N) ) ) ).

% INFM_nat_le
tff(fact_10210_ATP_Olambda__1,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ct,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_1
tff(fact_10211_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fr,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_2
tff(fact_10212_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_ge,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_3
tff(fact_10213_ATP_Olambda__4,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gj,Uu)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu))) ) ).

% ATP.lambda_4
tff(fact_10214_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_dp,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ca,Uu)) ).

% ATP.lambda_5
tff(fact_10215_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_ev,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bv,Uu)) ).

% ATP.lambda_6
tff(fact_10216_ATP_Olambda__7,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_fa,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_ez(Uu)))) ).

% ATP.lambda_7
tff(fact_10217_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_ey,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_ex(Uu)))) ).

% ATP.lambda_8
tff(fact_10218_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_cx,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_9
tff(fact_10219_ATP_Olambda__10,axiom,
    ! [Uu: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aTP_Lamm_gd,Uu)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Uu,N)) ) ) ) ).

% ATP.lambda_10
tff(fact_10220_ATP_Olambda__11,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_fg,Uu)
    <=> ? [I4: int,N: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N)) )
          & ( N != zero_zero_nat ) ) ) ).

% ATP.lambda_11
tff(fact_10221_ATP_Olambda__12,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      aa_nat_Extended_enat(aTP_Lamm_gt(Uu),Uua) = extend3600170679010898289d_enat(aTP_Lamm_gs(Uua),
        $ite(Uua = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),
        Uu) ).

% ATP.lambda_12
tff(fact_10222_ATP_Olambda__13,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_go(Uu),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(aTP_Lamm_bh,Uua),$false),Uu) ) ).

% ATP.lambda_13
tff(fact_10223_ATP_Olambda__14,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_gp(Uu),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),$true),Uu) ) ).

% ATP.lambda_14
tff(fact_10224_ATP_Olambda__15,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bz,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_15
tff(fact_10225_ATP_Olambda__16,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_cq,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_16
tff(fact_10226_ATP_Olambda__17,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_cp,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_17
tff(fact_10227_ATP_Olambda__18,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_dj,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_18
tff(fact_10228_ATP_Olambda__19,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_fi(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_19
tff(fact_10229_ATP_Olambda__20,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_cu(Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_20
tff(fact_10230_ATP_Olambda__21,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_cy(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_21
tff(fact_10231_ATP_Olambda__22,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ca,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_22
tff(fact_10232_ATP_Olambda__23,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dq,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_23
tff(fact_10233_ATP_Olambda__24,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_eh(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_24
tff(fact_10234_ATP_Olambda__25,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gk(Uu),Uua)
    <=> ( ( Uu = aa_int_rat(fract(product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Uua))
        & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua)) ) ) ).

% ATP.lambda_25
tff(fact_10235_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gl,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_26
tff(fact_10236_ATP_Olambda__27,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ff(Uu),Uua)
    <=> ( aa_set_int_o(member_int(Uua),Uu)
        & ! [X: int] :
            ( aa_set_int_o(member_int(X),Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Uua) ) ) ) ).

% ATP.lambda_27
tff(fact_10237_ATP_Olambda__28,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ci,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_28
tff(fact_10238_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_29
tff(fact_10239_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_br(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_30
tff(fact_10240_ATP_Olambda__31,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_by,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_31
tff(fact_10241_ATP_Olambda__32,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_cs,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_32
tff(fact_10242_ATP_Olambda__33,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ei(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_eh(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_33
tff(fact_10243_ATP_Olambda__34,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ep(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_eh(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_34
tff(fact_10244_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_do,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_35
tff(fact_10245_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cf,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_36
tff(fact_10246_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ck,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_37
tff(fact_10247_ATP_Olambda__38,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_dl,Uu),Uua) = $let(
        l2: num,
        l2:= 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_38
tff(fact_10248_ATP_Olambda__39,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_di,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_39
tff(fact_10249_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_gr(Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% ATP.lambda_40
tff(fact_10250_ATP_Olambda__41,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_dm(Uu),Uua) = groups6591440286371151544t_real(Uu,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_41
tff(fact_10251_ATP_Olambda__42,axiom,
    ! [Uu: set_Extended_enat,Uua: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(aTP_Lamm_aw(Uu),Uua)
    <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(Uua),Uu) ) ).

% ATP.lambda_42
tff(fact_10252_ATP_Olambda__43,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_av(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_43
tff(fact_10253_ATP_Olambda__44,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_au,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_44
tff(fact_10254_ATP_Olambda__45,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_ax(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_45
tff(fact_10255_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bh,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_46
tff(fact_10256_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_47
tff(fact_10257_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bg,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_48
tff(fact_10258_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ej,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_49
tff(fact_10259_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dk,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_50
tff(fact_10260_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_dr,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_51
tff(fact_10261_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_52
tff(fact_10262_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ds,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_53
tff(fact_10263_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bv,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_54
tff(fact_10264_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_em,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_55
tff(fact_10265_ATP_Olambda__56,axiom,
    ! [Uu: set_Extended_enat,Uua: extended_enat] :
      ( aa_Extended_enat_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set3337841544853333262enat_o(member_Extended_enat(Uua),Uu) ) ).

% ATP.lambda_56
tff(fact_10266_ATP_Olambda__57,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_57
tff(fact_10267_ATP_Olambda__58,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_58
tff(fact_10268_ATP_Olambda__59,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_aa(Uu),(Uua))
    <=> aa_set_o_o(member_o((Uua)),Uu) ) ).

% ATP.lambda_59
tff(fact_10269_ATP_Olambda__60,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_60
tff(fact_10270_ATP_Olambda__61,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_61
tff(fact_10271_ATP_Olambda__62,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_af(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_62
tff(fact_10272_ATP_Olambda__63,axiom,
    ! [Uu: extended_enat,Uua: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(aTP_Lamm_bf,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_63
tff(fact_10273_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_be,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_64
tff(fact_10274_ATP_Olambda__65,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_az(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_65
tff(fact_10275_ATP_Olambda__66,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_bc,(Uu)),(Uua))
    <=> ( (Uua)
      <=> (Uu) ) ) ).

% ATP.lambda_66
tff(fact_10276_ATP_Olambda__67,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_bd,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_67
tff(fact_10277_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ba,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_68
tff(fact_10278_ATP_Olambda__69,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bb,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_69
tff(fact_10279_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_gi(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_70
tff(fact_10280_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_cz(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_71
tff(fact_10281_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_fp,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_fo(Uu),Uua)) ).

% ATP.lambda_72
tff(fact_10282_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_fn,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_fm(Uu),Uua)) ).

% ATP.lambda_73
tff(fact_10283_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_ee(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ed(Uu),Uua)) ).

% ATP.lambda_74
tff(fact_10284_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_fh(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_75
tff(fact_10285_ATP_Olambda__76,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_gg(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_76
tff(fact_10286_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eo,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_77
tff(fact_10287_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_gs(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_78
tff(fact_10288_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gc,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_79
tff(fact_10289_ATP_Olambda__80,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gb,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_80
tff(fact_10290_ATP_Olambda__81,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ga,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_81
tff(fact_10291_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fz,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_82
tff(fact_10292_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_el(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_83
tff(fact_10293_ATP_Olambda__84,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_eu(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_84
tff(fact_10294_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cb(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_85
tff(fact_10295_ATP_Olambda__86,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_eg(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_86
tff(fact_10296_ATP_Olambda__87,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_gh(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Uu)) ).

% ATP.lambda_87
tff(fact_10297_ATP_Olambda__88,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_fe(Uu),Uua)
    <=> ! [X: real] :
          ( aa_set_real_o(member_real(X),Uu)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Uua) ) ) ).

% ATP.lambda_88
tff(fact_10298_ATP_Olambda__89,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dd(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_89
tff(fact_10299_ATP_Olambda__90,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_cv(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_90
tff(fact_10300_ATP_Olambda__91,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_cw(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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_91
tff(fact_10301_ATP_Olambda__92,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dh(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_92
tff(fact_10302_ATP_Olambda__93,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dg(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_93
tff(fact_10303_ATP_Olambda__94,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ef(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_94
tff(fact_10304_ATP_Olambda__95,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_df(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_95
tff(fact_10305_ATP_Olambda__96,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fx(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_fw(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_96
tff(fact_10306_ATP_Olambda__97,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fv(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_fu(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_97
tff(fact_10307_ATP_Olambda__98,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dv(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_98
tff(fact_10308_ATP_Olambda__99,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_dw(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_99
tff(fact_10309_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ec(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_100
tff(fact_10310_ATP_Olambda__101,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ch(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_101
tff(fact_10311_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_fk(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_102
tff(fact_10312_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_fl(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_103
tff(fact_10313_ATP_Olambda__104,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_fj(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_104
tff(fact_10314_ATP_Olambda__105,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bx(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_105
tff(fact_10315_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( finite_card_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_106
tff(fact_10316_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fs(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_107
tff(fact_10317_ATP_Olambda__108,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bw(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_108
tff(fact_10318_ATP_Olambda__109,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cl(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_109
tff(fact_10319_ATP_Olambda__110,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_db(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_110
tff(fact_10320_ATP_Olambda__111,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_111
tff(fact_10321_ATP_Olambda__112,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_da(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_112
tff(fact_10322_ATP_Olambda__113,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bl(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_113
tff(fact_10323_ATP_Olambda__114,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bk(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_114
tff(fact_10324_ATP_Olambda__115,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bj(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_115
tff(fact_10325_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gm(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_116
tff(fact_10326_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ew(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_117
tff(fact_10327_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gf(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_118
tff(fact_10328_ATP_Olambda__119,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ex(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_119
tff(fact_10329_ATP_Olambda__120,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ez(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu) ) ).

% ATP.lambda_120
tff(fact_10330_ATP_Olambda__121,axiom,
    ! [Uu: set_Pr4811707699266497531nteger,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> member157494554546826820nteger(produc1086072967326762835nteger(Uua,Uub),Uu) ) ).

% ATP.lambda_121
tff(fact_10331_ATP_Olambda__122,axiom,
    ! [Uu: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_bn(Uu),Uua),(Uub))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu) ) ).

% ATP.lambda_122
tff(fact_10332_ATP_Olambda__123,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_123
tff(fact_10333_ATP_Olambda__124,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_124
tff(fact_10334_ATP_Olambda__125,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(product_Pair_int_int(Uua,Uub),Uu) ) ).

% ATP.lambda_125
tff(fact_10335_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_126
tff(fact_10336_ATP_Olambda__127,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_127
tff(fact_10337_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ed(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_128
tff(fact_10338_ATP_Olambda__129,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cc(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_129
tff(fact_10339_ATP_Olambda__130,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cm(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_130
tff(fact_10340_ATP_Olambda__131,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_131
tff(fact_10341_ATP_Olambda__132,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ft(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_132
tff(fact_10342_ATP_Olambda__133,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dc(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_133
tff(fact_10343_ATP_Olambda__134,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ek(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_134
tff(fact_10344_ATP_Olambda__135,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_du(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_135
tff(fact_10345_ATP_Olambda__136,axiom,
    ! [Uu: fun_Extended_enat_o,Uua: fun_Extended_enat_o,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_fun3141604562678283816enat_o(aTP_Lamm_am(Uu),Uua),Uub)
    <=> ( aa_Extended_enat_o(Uu,Uub)
        | aa_Extended_enat_o(Uua,Uub) ) ) ).

% ATP.lambda_136
tff(fact_10346_ATP_Olambda__137,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_137
tff(fact_10347_ATP_Olambda__138,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        | aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_138
tff(fact_10348_ATP_Olambda__139,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ah(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_139
tff(fact_10349_ATP_Olambda__140,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_140
tff(fact_10350_ATP_Olambda__141,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_141
tff(fact_10351_ATP_Olambda__142,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_142
tff(fact_10352_ATP_Olambda__143,axiom,
    ! [Uu: fun_Extended_enat_o,Uua: fun_Extended_enat_o,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_fun3141604562678283816enat_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( aa_Extended_enat_o(Uu,Uub)
        & aa_Extended_enat_o(Uua,Uub) ) ) ).

% ATP.lambda_143
tff(fact_10353_ATP_Olambda__144,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_144
tff(fact_10354_ATP_Olambda__145,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_145
tff(fact_10355_ATP_Olambda__146,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_146
tff(fact_10356_ATP_Olambda__147,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_147
tff(fact_10357_ATP_Olambda__148,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_148
tff(fact_10358_ATP_Olambda__149,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_149
tff(fact_10359_ATP_Olambda__150,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_150
tff(fact_10360_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_eq(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_151
tff(fact_10361_ATP_Olambda__152,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_dt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_152
tff(fact_10362_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_en(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_153
tff(fact_10363_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ce(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_154
tff(fact_10364_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_et(Uu),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_155
tff(fact_10365_ATP_Olambda__156,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_gq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Uu),Uua) ).

% ATP.lambda_156
tff(fact_10366_ATP_Olambda__157,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gn(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_157
tff(fact_10367_ATP_Olambda__158,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_er(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_158
tff(fact_10368_ATP_Olambda__159,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_fc(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_159
tff(fact_10369_ATP_Olambda__160,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_co(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_cn(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_160
tff(fact_10370_ATP_Olambda__161,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_eb(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_161
tff(fact_10371_ATP_Olambda__162,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_dz(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_162
tff(fact_10372_ATP_Olambda__163,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_dy(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_163
tff(fact_10373_ATP_Olambda__164,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_dx(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_164
tff(fact_10374_ATP_Olambda__165,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_fu(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_165
tff(fact_10375_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_fm(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_166
tff(fact_10376_ATP_Olambda__167,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_fw(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_167
tff(fact_10377_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_fo(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_168
tff(fact_10378_ATP_Olambda__169,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_cn(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_169
tff(fact_10379_ATP_Olambda__170,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_ea(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_dz(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_170
tff(fact_10380_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_fd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I4)),X_12)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_171
tff(fact_10381_ATP_Olambda__172,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cg,Uu) = Uu ).

% ATP.lambda_172
tff(fact_10382_ATP_Olambda__173,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cd,Uu) = Uu ).

% ATP.lambda_173
tff(fact_10383_ATP_Olambda__174,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cj,Uu) = Uu ).

% ATP.lambda_174
tff(fact_10384_ATP_Olambda__175,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_cr,Uu)
    <=> $false ) ).

% ATP.lambda_175
tff(fact_10385_ATP_Olambda__176,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_es,Uu)
    <=> $false ) ).

% ATP.lambda_176
tff(fact_10386_ATP_Olambda__177,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_fy,Uu)
    <=> $true ) ).

% ATP.lambda_177

% Helper facts (15)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y6: $o] :
      ( ( (X8) != (Y6) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y6)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y6: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y6))
      | ( (X8) = (Y6) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y6: int] :
      ( ( X8 != Y6 )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y6: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y6)
      | ( X8 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y6: nat] :
      ( ( X8 != Y6 )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y6: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y6)
      | ( X8 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X8: real,Y6: real] :
      ( ( X8 != Y6 )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X8),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X8: real,Y6: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X8),Y6)
      | ( X8 = Y6 ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X6: real] : aa_real_o(P,X6) ) ) ).

tff(help_fequal_2_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X8: extended_enat,Y6: extended_enat] :
      ( ( X8 != Y6 )
      | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(fequal_Extended_enat,X8),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X8: extended_enat,Y6: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(fequal_Extended_enat,X8),Y6)
      | ( X8 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X8: set_nat,Y6: set_nat] :
      ( ( X8 != Y6 )
      | aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X8),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X8: set_nat,Y6: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X8),Y6)
      | ( X8 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X8: list_nat,Y6: list_nat] :
      ( ( X8 != Y6 )
      | aa_list_nat_o(fequal_list_nat(X8),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X8: list_nat,Y6: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X8),Y6)
      | ( X8 = Y6 ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_set_nat_o(member_nat(y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_VEBT_set_vebt(t)),insert_nat(x,bot_bot_set_nat))) ).

%------------------------------------------------------------------------------