%------------------------------------------------------------------------------
% File     : ITP278_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00535_035038
% 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    : 0075_VEBT_Uniqueness_00535_035038 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.33 v8.1.0
% Syntax   : Number of formulae    : 13600 (4408 unt;2698 typ;   0 def)
%            Number of atoms       : 26887 (9645 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17074 (2504   ~; 467   |;1265   &)
%                                         (2643 <=>;10195  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       : 2559 (1126 fml;1433 var)
%            Number of X terms     :  958 (   0  []; 775 ite; 183 let)
%            Number of types       :  510 ( 509 usr)
%            Number of type conns  : 2580 (1684   >; 896   *;   0   +;   0  <<)
%            Number of predicates  :  140 ( 137 usr;   3 prp; 0-4 aty)
%            Number of functors    : 2074 (2074 usr; 526 con; 0-8 aty)
%            Number of variables   : 27285 (26681   !; 604   ?;27285   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 16:35:28.943
%------------------------------------------------------------------------------
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    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 (2189)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_real > fun_real_o ).

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

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_nat > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: produc3843707927480180839at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_list_nat > fun_se1865396762483378772_nat_o ).

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

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_Pr4329608150637261639at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_Pr4329608150637261639at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_Pr4329608150637261639at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_Pr1261947904930325089at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: set_list_nat > fun_fu1994797156054922755_nat_o ).

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

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_complex_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_o_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_int_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: produc3843707927480180839at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: $o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: product_prod_nat_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: $o > fun_fun_o_o_fun_o_o ).

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

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

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

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_nat > fun_set_nat_o ).

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

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_o > fun_fu2389858412225411757un_o_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_o > fun_fu8756204934394872281un_o_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_o > fun_fu6337961365400742745un_o_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_o > fun_fu2641566564518291801un_o_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_complex > fun_fu2497916604042137945plex_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_complex > fun_fu6925816717872577647plex_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_o_o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_o_nat > fun_fu2310066269493208999at_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_o > fun_fu3079103113578437785_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_nat > fun_fu1217771548038759377at_nat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_o_3115027942823555909_nat_o > fun_fu8040060317109070833_nat_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_na3970208099681795815_nat_o > fun_fu484727072080742171_nat_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_fu6891447142836604759at_o_o > fun_fu8398852373182072911_nat_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_fu1115759055147092071_o_nat > fun_fu4228150600790569371at_nat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_o_4034989691794805934at_nat > fun_fu1874366598054176346at_nat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_Pr4796943543331071514_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_Pr2602117396471556196at_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Co2470855966376689427nteger ).

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

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

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_Pr4796943543331071514_nat_o > fun_Pr4796943543331071514_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: $o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: real > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_real > fun_fu6899955116716589935t_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_complex_o > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_complex_o > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_complex_o > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_complex_o > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_o > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_nat_o > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_o > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_complex_o > fun_fu6919794090103715845omplex ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_o > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_int_o > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_co8153928105989009288nteger > fun_Co8416587583108374242nteger ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_co8153928105989009288nteger > fun_Co8975588572093281791plex_o ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_co8153928105989009288nteger > fun_Co8975588572093281791plex_o ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_complex_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_complex_nat > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_complex_nat > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_complex_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_complex_int > fun_in5065666251602127472plex_o ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_complex_int > fun_in5065666251602127472plex_o ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_se7586849273513198688nteger > fun_Co4664883906889918906nteger ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_se7586849273513198688nteger > fun_Co4326030104980677927_nat_o ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_se7586849273513198688nteger > fun_Co4326030104980677927_nat_o ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_li3231056112175589690nteger > fun_Co4193813265255946772nteger ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_li3231056112175589690nteger > fun_Co8418209992770183757_nat_o ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_li3231056112175589690nteger > fun_Co8418209992770183757_nat_o ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_set_nat_nat > fun_na106483476154090400at_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_set_nat_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_set_nat_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: $o > fun_fu2803337334652888941o_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: $o > fun_fu9039538216874201369_o_rat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_o_Code_integer > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_o_real > fun_fu8103376058317145219o_real ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_int > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_int_int ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_real ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: real > fun_fu4063443379060642216t_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_real_real ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_int_int ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_nat_complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_real > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_real_o ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_o ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_real_real ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_real_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_nat_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: real > fun_re8658731967611918585t_real ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__zy____,type,
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tff(sy_c_ATP_058Lamm__zz____,type,
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tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,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_Oround_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
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tff(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
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tff(sy_c_Binomial_Obinomial,type,
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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    sup_su7393609867835711046_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_int_fun_int_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    sup_su2398265404452057824_nat_o: ( fun_Pr3059359313727237556_nat_o * fun_Pr3059359313727237556_nat_o ) > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    sup_su5424986631308460290_nat_o: ( fun_se5639961202347202350_nat_o * fun_se5639961202347202350_nat_o ) > fun_se5639961202347202350_nat_o ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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tff(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_List_Oappend_001t__Nat__Onat,type,
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tff(sy_c_List_Odrop_001t__Nat__Onat,type,
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tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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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,
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tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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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_Otl_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001_Eo,type,
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tff(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
    list_update_complex: ( list_complex * nat * complex ) > list_complex ).

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,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
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tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Onth_001_Eo,type,
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tff(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
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tff(sy_c_List_Onth_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Num__Onum,type,
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tff(sy_c_List_Oreplicate_001_Eo,type,
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tff(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_I_Eo_M_Eo_J,type,
    ord_less_fun_o_o: ( fun_o_o * fun_o_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le4409593196786536943plex_o: ( fun_complex_o * fun_complex_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_less_fun_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_M_Eo_J,type,
    ord_less_fun_real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
    ord_less_eq_o: fun_o_fun_o_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: set_o > fun_set_o_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: ( set_list_nat * set_list_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > fun_se7788542033685991124_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__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_M_Eo_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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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le3306795289226699318_num_o: ( fun_num_fun_num_o * fun_num_fun_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,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,
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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,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: fun_se1909708154429037836et_nat ).

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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_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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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__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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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 > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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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,
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tff(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: ( $o * $o ) > product_prod_o_o ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: ( $o * int ) > product_prod_o_int ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: ( $o * nat ) > product_prod_o_nat ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

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

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: ( option4927543243414619207at_nat * option4927543243414619207at_nat ) > produc6121120109295599847at_nat ).

tff(sy_c_Product__Type_OPair_001t__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__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Ounit_ORep__unit,type,
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tff(sy_c_Real_Ocauchy,type,
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tff(sy_c_Real_Orealrel,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__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,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__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__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__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__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,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_It__Nat__Onat_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__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__Int__Oint,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__Real__Oreal,type,
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tff(sy_c_Set_Othe__elem_001_Eo,type,
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tff(sy_c_Set_Othe__elem_001t__Int__Oint,type,
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tff(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
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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__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat * nat ) > fun_complex_complex ).

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat ) > fun_rat_rat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
    set_or8904488021354931149Most_o: ( $o * $o ) > set_o ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
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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__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
    set_ord_atMost_o: $o > set_o ).

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_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > fun_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_001_Eo,type,
    set_ord_lessThan_o: $o > set_o ).

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__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

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_Oascii__of,type,
    ascii_of: char > char ).

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

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

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
    topolo4899668324122417113eq_int: fun_nat_int > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: fun_nat_nat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: fun_nat_num > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: fun_nat_rat > $o ).

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: fun_nat_set_nat > $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__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: fun_nat_complex > fun_nat_complex ).

tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: fun_nat_int > fun_nat_int ).

tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

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

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

tff(sy_c_Transcendental_Opowr__real,type,
    powr_real2: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__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_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
    type_d6188575255521822967unit_o: ( fun_Product_unit_o * fun_o_Product_unit ) > fun_set_o_o ).

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__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__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_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( fun_num_fun_num_num * option_num * option_num ) > option_num ).

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    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
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tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li1025265955262268170_nat_o > fun_list_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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tff(sy_c_Wellfounded_Oless__than,type,
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tff(sy_c_aa_001_Eo_001_Eo,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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    aa_fun2566793329686429606_nat_o: ( fun_fu1322491050725084261_nat_o * fun_Pr3059359313727237556_nat_o ) > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_aa_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J_001_Eo,type,
    aa_fun_rat_rat_o: ( fun_fun_rat_rat_o * fun_rat_rat ) > $o ).

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun7923231501384339402nteger: ( fun_fu8643405727515436425nteger * fun_re6978948838062957830nteger ) > fun_re6978948838062957830nteger ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Int__Oint_J_001t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    aa_fun8270430325945277160al_int: ( fun_fu3737244727311182119al_int * fun_real_int ) > fun_real_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Int__Oint_J_001t__fun_It__fun_It__Real__Oreal_Mt__Int__Oint_J_Mt__fun_It__Real__Oreal_Mt__Int__Oint_J_J,type,
    aa_fun7818883909550569274al_int: ( fun_fu5645595436833112697al_int * fun_real_int ) > fun_fu3737244727311182119al_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
    aa_fun211602392931233840al_nat: ( fun_fu3214020956574444655al_nat * fun_real_nat ) > fun_real_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Nat__Onat_J_Mt__fun_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    aa_fun1500080954902701606al_nat: ( fun_fu3768230810189712741al_nat * fun_real_nat ) > fun_fu3214020956574444655al_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun3209485225100452678real_o: ( fun_fu6510078244439976541real_o * fun_real_rat ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
    aa_fun2468082234287226944al_rat: ( fun_fu2598743018846156927al_rat * fun_real_rat ) > fun_real_rat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Rat__Orat_J_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun8459995508950063326real_o: ( fun_fu5587483295140630005real_o * fun_real_rat ) > fun_fu6510078244439976541real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Rat__Orat_J_Mt__fun_It__Real__Oreal_Mt__Rat__Orat_J_J,type,
    aa_fun1537150169341160510al_rat: ( fun_fu7187499828406537085al_rat * fun_real_rat ) > fun_fu2598743018846156927al_rat ).

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

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun3693225987673163702real_o: ( fun_fu5533088652784741581real_o * fun_real_real ) > fun_fu3022329567491847689real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    aa_fun5414376518349398074l_real: ( fun_fu9132278742410140537l_real * fun_real_real ) > fun_fu3077872570260440615l_real ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_fun751667557118977660_nat_o: ( fun_fu2408719596384163971_nat_o * fun_set_nat_o ) > fun_set_nat_o ).

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1148497455491383662plex_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__List__Olist_It__Nat__Onat_J,type,
    fequal_list_nat: list_nat > fun_list_nat_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_o ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fequal9188845342289841878at_nat: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fequal7949547063978738364at_nat: produc3843707927480180839at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_fequal_001t__Set__Oset_It__Nat__Onat_J,type,
    fequal_set_nat: set_nat > fun_set_nat_o ).

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

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

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

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

tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $o ).

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

tff(sy_c_member_001t__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__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 > fun_se4782206882984618170_nat_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).

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

tff(sy_c_member_001t__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__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: ( produc2934264451710624999od_o_o * set_Pr1932065953672099015od_o_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: ( produc732395585841259969at_nat * set_Pr457366540195662369at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > fun_se7788542033685991124_nat_o ).

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

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

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

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

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_info____,type,
    info: option4927543243414619207at_nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_sa____,type,
    sa: vEBT_VEBT ).

tff(sy_v_summary_H____,type,
    summary: vEBT_VEBT ).

tff(sy_v_summary____,type,
    summary2: vEBT_VEBT ).

tff(sy_v_treeList_H____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList2: list_VEBT_VEBT ).

% Relevant facts (10883)
tff(fact_0__092_060open_062summary_A_061_Asummary_H_092_060close_062,axiom,
    summary2 = summary ).

% \<open>summary = summary'\<close>
tff(fact_1_case4_I9_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% case4(9)
tff(fact_2_infsplit,axiom,
    info = some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)) ).

% infsplit
tff(fact_3_case4_I13_J,axiom,
    vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2)) = vEBT_VEBT_set_vebt(sa) ).

% case4(13)
tff(fact_4_case4_I12_J,axiom,
    aa_nat_o(vEBT_invar_vebt(sa),deg) ).

% case4(12)
tff(fact_5_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_6_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_7_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( some_nat(X2) = some_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_8_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_9__092_060open_062treeList_A_061_AtreeList_H_092_060close_062,axiom,
    treeList2 = treeList ).

% \<open>treeList = treeList'\<close>
tff(fact_10_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(X1,X2) = produc2922128104949294807at_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_11_prod_Oinject,axiom,
    ! [X1: num,X2: num,Y1: num,Y2: num] :
      ( ( product_Pair_num_num(X1,X2) = product_Pair_num_num(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_12_prod_Oinject,axiom,
    ! [X1: nat,X2: num,Y1: nat,Y2: num] :
      ( ( product_Pair_nat_num(X1,X2) = product_Pair_nat_num(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_13_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_14_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_15_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(A,B) = produc2922128104949294807at_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_16_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_17_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_18_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_19_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_20_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList2))
       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ).

% case4(8)
tff(fact_21__092_060open_062set__vebt_H_Asummary_A_061_Aset__vebt_H_Asummary_H_092_060close_062,axiom,
    vEBT_VEBT_set_vebt(summary2) = vEBT_VEBT_set_vebt(summary) ).

% \<open>set_vebt' summary = set_vebt' summary'\<close>
tff(fact_22__092_060open_062_092_060And_062x_O_Avebt__member_Asummary_Ax_A_061_Avebt__member_Asummary_H_Ax_092_060close_062,axiom,
    ! [X3: nat] :
      ( aa_nat_o(vEBT_vebt_member(summary2),X3)
    <=> aa_nat_o(vEBT_vebt_member(summary),X3) ) ).

% \<open>\<And>x. vebt_member summary x = vebt_member summary' x\<close>
tff(fact_23_aa,axiom,
    aa_set_nat_o(ord_less_eq_set_nat(insert_nat(mi,insert_nat(ma,bot_bot_set_nat))),vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2))) ).

% aa
tff(fact_24_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] : X3 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),M) ).

% prod_decode_aux.cases
tff(fact_25_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : Y != produc2922128104949294807at_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_26_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A3: num,B3: num] : Y != product_Pair_num_num(A3,B3) ).

% old.prod.exhaust
tff(fact_27_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_num] :
      ~ ! [A3: nat,B3: num] : Y != product_Pair_nat_num(A3,B3) ).

% old.prod.exhaust
tff(fact_28_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) ).

% old.prod.exhaust
tff(fact_29_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) ).

% old.prod.exhaust
tff(fact_30_deg__deg__n,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_31_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X3)
      <=> aa_nat_o(vEBT_vebt_member(Ta),X3) ) ) ).

% both_member_options_equiv_member
tff(fact_32_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X3)
       => aa_nat_o(vEBT_vebt_member(Ta),X3) ) ) ).

% valid_member_both_member_options
tff(fact_33_max__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),X3)
    <=> ( aa_set_nat_o(member_nat(X3),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),X3) ) ) ) ).

% max_in_set_def
tff(fact_34_min__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),X3)
    <=> ( aa_set_nat_o(member_nat(X3),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),X4) ) ) ) ).

% min_in_set_def
tff(fact_35_insert_H__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_VEBT_insert(Ta,X3)),Nb) ) ).

% insert'_pres_valid
tff(fact_36_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) )
          & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_37_lesseq__shift,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(X3)),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_38_case4_I3_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary2),m) ).

% case4(3)
tff(fact_39_case4_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList2))
     => ( aa_nat_o(vEBT_invar_vebt(X),na)
        & ! [Xa: vEBT_VEBT] :
            ( aa_nat_o(vEBT_invar_vebt(Xa),na)
           => ( ( vEBT_VEBT_set_vebt(X) = vEBT_VEBT_set_vebt(Xa) )
             => ( Xa = X ) ) ) ) ) ).

% case4(1)
tff(fact_40_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),X3)
      <=> aa_set_nat_o(member_nat(X3),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_41_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_42_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(A,B) = produc2922128104949294807at_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_43_Pair__inject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_44_Pair__inject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_45_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_46_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_47_prod__cases,axiom,
    ! [P: fun_Pr3059359313727237556_nat_o,P2: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_Pro5947980084102686253_nat_o(P,produc2922128104949294807at_nat(A3,B3))
     => aa_Pro5947980084102686253_nat_o(P,P2) ) ).

% prod_cases
tff(fact_48_prod__cases,axiom,
    ! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
      ( ! [A3: num,B3: num] : aa_Pro7685007914727650815_num_o(P,product_Pair_num_num(A3,B3))
     => aa_Pro7685007914727650815_num_o(P,P2) ) ).

% prod_cases
tff(fact_49_prod__cases,axiom,
    ! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
      ( ! [A3: nat,B3: num] : aa_Pro6908818167415475913_num_o(P,product_Pair_nat_num(A3,B3))
     => aa_Pro6908818167415475913_num_o(P,P2) ) ).

% prod_cases
tff(fact_50_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_51_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A3: int,B3: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_52_surj__pair,axiom,
    ! [P2: produc3843707927480180839at_nat] :
    ? [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : P2 = produc2922128104949294807at_nat(X5,Y3) ).

% surj_pair
tff(fact_53_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X5: num,Y3: num] : P2 = product_Pair_num_num(X5,Y3) ).

% surj_pair
tff(fact_54_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X5: nat,Y3: num] : P2 = product_Pair_nat_num(X5,Y3) ).

% surj_pair
tff(fact_55_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X5: nat,Y3: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) ).

% surj_pair
tff(fact_56_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X5: int,Y3: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Y3) ).

% surj_pair
tff(fact_57_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A,A4) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_58_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A,A4) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_59_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( insert_o((B),bot_bot_set_o) = insert_o((A),A4) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A4),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_60_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A4) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A4),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_61_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A4) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_62_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,A4) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_63_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,A4) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_64_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( insert_o((A),A4) = insert_o((B),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A4),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_65_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( insert_int(A,A4) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A4),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_66_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( insert_nat(A,A4) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_67_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_68_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_69_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_70_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_71_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_72_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_73_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),collec3392354462482085612at_nat(P))
    <=> aa_Pro6150637604445964947_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_74_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_75_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_76_Collect__mem__eq,axiom,
    ! [A4: set_real] : collect_real(aTP_Lamm_a(A4)) = A4 ).

% Collect_mem_eq
tff(fact_77_Collect__mem__eq,axiom,
    ! [A4: set_o] : collect_o(aTP_Lamm_aa(A4)) = A4 ).

% Collect_mem_eq
tff(fact_78_Collect__mem__eq,axiom,
    ! [A4: set_int] : collect_int(aTP_Lamm_ab(A4)) = A4 ).

% Collect_mem_eq
tff(fact_79_Collect__mem__eq,axiom,
    ! [A4: set_nat] : collect_nat(aTP_Lamm_ac(A4)) = A4 ).

% Collect_mem_eq
tff(fact_80_Collect__mem__eq,axiom,
    ! [A4: set_complex] : collect_complex(aTP_Lamm_ad(A4)) = A4 ).

% Collect_mem_eq
tff(fact_81_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_82_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : collect_set_nat(aTP_Lamm_af(A4)) = A4 ).

% Collect_mem_eq
tff(fact_83_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : collect_list_nat(aTP_Lamm_ag(A4)) = A4 ).

% Collect_mem_eq
tff(fact_84_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_nat_o(P,X5)
        <=> aa_nat_o(Q,X5) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_85_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X5: complex] :
          ( aa_complex_o(P,X5)
        <=> aa_complex_o(Q,X5) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_86_Collect__cong,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ! [X5: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P,X5)
        <=> aa_Pro6150637604445964947_nat_o(Q,X5) )
     => ( collec3392354462482085612at_nat(P) = collec3392354462482085612at_nat(Q) ) ) ).

% Collect_cong
tff(fact_87_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X5: set_nat] :
          ( aa_set_nat_o(P,X5)
        <=> aa_set_nat_o(Q,X5) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_88_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X5: list_nat] :
          ( aa_list_nat_o(P,X5)
        <=> aa_list_nat_o(Q,X5) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_89_insert__subset,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(X3,A4)),B4)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B4)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ) ).

% insert_subset
tff(fact_90_insert__subset,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X3,A4)),B4)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),B4)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4) ) ) ).

% insert_subset
tff(fact_91_insert__subset,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(insert_complex(X3,A4)),B4)
    <=> ( aa_set_complex_o(member_complex(X3),B4)
        & aa_set_complex_o(ord_le211207098394363844omplex(A4),B4) ) ) ).

% insert_subset
tff(fact_92_insert__subset,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(X3,A4)),B4)
    <=> ( aa_set_real_o(member_real(X3),B4)
        & aa_set_real_o(ord_less_eq_set_real(A4),B4) ) ) ).

% insert_subset
tff(fact_93_insert__subset,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o((X3),A4)),B4)
    <=> ( aa_set_o_o(member_o((X3)),B4)
        & aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ) ).

% insert_subset
tff(fact_94_insert__subset,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(X3,A4)),B4)
    <=> ( aa_set_int_o(member_int(X3),B4)
        & aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ) ).

% insert_subset
tff(fact_95_insert__subset,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(X3,A4)),B4)
    <=> ( aa_set_nat_o(member_nat(X3),B4)
        & aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ) ).

% insert_subset
tff(fact_96_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% singletonI
tff(fact_97_singletonI,axiom,
    ! [A: complex] : aa_set_complex_o(member_complex(A),insert_complex(A,bot_bot_set_complex)) ).

% singletonI
tff(fact_98_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real(A),insert_real(A,bot_bot_set_real)) ).

% singletonI
tff(fact_99_singletonI,axiom,
    ! [A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% singletonI
tff(fact_100_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),insert_o((A),bot_bot_set_o)) ).

% singletonI
tff(fact_101_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),insert_nat(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_102_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_103_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),bot_bo2099793752762293965at_nat)
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_104_subset__empty,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),bot_bot_set_o)
    <=> ( A4 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_105_subset__empty,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),bot_bot_set_int)
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_106_subset__empty,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),bot_bot_set_nat)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_107_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A4) ).

% empty_subsetI
tff(fact_108_empty__subsetI,axiom,
    ! [A4: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A4) ).

% empty_subsetI
tff(fact_109_empty__subsetI,axiom,
    ! [A4: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A4) ).

% empty_subsetI
tff(fact_110_empty__subsetI,axiom,
    ! [A4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A4) ).

% empty_subsetI
tff(fact_111_case4_I2_J,axiom,
    ! [Sb: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Sb),m)
     => ( ( vEBT_VEBT_set_vebt(summary2) = vEBT_VEBT_set_vebt(Sb) )
       => ( Sb = summary2 ) ) ) ).

% case4(2)
tff(fact_112_aca,axiom,
    size_s6755466524823107622T_VEBT(treeList) = size_s6755466524823107622T_VEBT(treeList2) ).

% aca
tff(fact_113_subset__singletonD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))
     => ( ( A4 = bot_bo228742789529271731at_nat )
        | ( A4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_114_subset__singletonD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))
     => ( ( A4 = bot_bo2099793752762293965at_nat )
        | ( A4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_115_subset__singletonD,axiom,
    ! [A4: set_o,X3: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X3),bot_bot_set_o))
     => ( ( A4 = bot_bot_set_o )
        | ( A4 = insert_o((X3),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_116_subset__singletonD,axiom,
    ! [A4: set_int,X3: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),insert_int(X3,bot_bot_set_int))
     => ( ( A4 = bot_bot_set_int )
        | ( A4 = insert_int(X3,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_117_subset__singletonD,axiom,
    ! [A4: set_nat,X3: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(X3,bot_bot_set_nat))
     => ( ( A4 = bot_bot_set_nat )
        | ( A4 = insert_nat(X3,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_118_subset__singleton__iff,axiom,
    ! [X6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X6),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( ( X6 = bot_bo228742789529271731at_nat )
        | ( X6 = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_119_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X6),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6 = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_120_subset__singleton__iff,axiom,
    ! [X6: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X6),insert_o((A),bot_bot_set_o))
    <=> ( ( X6 = bot_bot_set_o )
        | ( X6 = insert_o((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_121_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X6),insert_int(A,bot_bot_set_int))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_122_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X6),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_123_empty__Collect__eq,axiom,
    ! [P: fun_complex_o] :
      ( ( bot_bot_set_complex = collect_complex(P) )
    <=> ! [X4: complex] : ~ aa_complex_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_124_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_125_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_126_empty__Collect__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(P) )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_127_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P) )
    <=> ! [X4: $o] : ~ aa_o_o(P,(X4)) ) ).

% empty_Collect_eq
tff(fact_128_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_129_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_130_Collect__empty__eq,axiom,
    ! [P: fun_complex_o] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ! [X4: complex] : ~ aa_complex_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_131_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_132_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_133_Collect__empty__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_134_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ! [X4: $o] : ~ aa_o_o(P,(X4)) ) ).

% Collect_empty_eq
tff(fact_135_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_136_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_137_all__not__in__conv,axiom,
    ! [A4: set_complex] :
      ( ! [X4: complex] : ~ aa_set_complex_o(member_complex(X4),A4)
    <=> ( A4 = bot_bot_set_complex ) ) ).

% all_not_in_conv
tff(fact_138_all__not__in__conv,axiom,
    ! [A4: set_real] :
      ( ! [X4: real] : ~ aa_set_real_o(member_real(X4),A4)
    <=> ( A4 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_139_all__not__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_140_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ! [X4: $o] : ~ aa_set_o_o(member_o((X4)),A4)
    <=> ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_141_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A4)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_142_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A4)
    <=> ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_143_empty__iff,axiom,
    ! [C: complex] : ~ aa_set_complex_o(member_complex(C),bot_bot_set_complex) ).

% empty_iff
tff(fact_144_empty__iff,axiom,
    ! [C: real] : ~ aa_set_real_o(member_real(C),bot_bot_set_real) ).

% empty_iff
tff(fact_145_empty__iff,axiom,
    ! [C: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),bot_bo2099793752762293965at_nat) ).

% empty_iff
tff(fact_146_empty__iff,axiom,
    ! [C: $o] : ~ aa_set_o_o(member_o((C)),bot_bot_set_o) ).

% empty_iff
tff(fact_147_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_148_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_149_subset__antisym,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( A4 = B4 ) ) ) ).

% subset_antisym
tff(fact_150_subsetI,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_set_complex_o(member_complex(X5),B4) )
     => aa_set_complex_o(ord_le211207098394363844omplex(A4),B4) ) ).

% subsetI
tff(fact_151_subsetI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_set_real_o(member_real(X5),B4) )
     => aa_set_real_o(ord_less_eq_set_real(A4),B4) ) ).

% subsetI
tff(fact_152_subsetI,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_set_o_o(member_o((X5)),B4) )
     => aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ).

% subsetI
tff(fact_153_subsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_set_int_o(member_int(X5),B4) )
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% subsetI
tff(fact_154_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_set_nat_o(member_nat(X5),B4) )
     => aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ).

% subsetI
tff(fact_155_insert__absorb2,axiom,
    ! [X3: nat,A4: set_nat] : insert_nat(X3,insert_nat(X3,A4)) = insert_nat(X3,A4) ).

% insert_absorb2
tff(fact_156_insert__absorb2,axiom,
    ! [X3: int,A4: set_int] : insert_int(X3,insert_int(X3,A4)) = insert_int(X3,A4) ).

% insert_absorb2
tff(fact_157_insert__absorb2,axiom,
    ! [X3: $o,A4: set_o] : insert_o((X3),insert_o((X3),A4)) = insert_o((X3),A4) ).

% insert_absorb2
tff(fact_158_insert__absorb2,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X3,insert8211810215607154385at_nat(X3,A4)) = insert8211810215607154385at_nat(X3,A4) ).

% insert_absorb2
tff(fact_159_insert__absorb2,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X3,insert9069300056098147895at_nat(X3,A4)) = insert9069300056098147895at_nat(X3,A4) ).

% insert_absorb2
tff(fact_160_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A4))
    <=> ( ( A = B )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4) ) ) ).

% insert_iff
tff(fact_161_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A4))
    <=> ( ( A = B )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4) ) ) ).

% insert_iff
tff(fact_162_insert__iff,axiom,
    ! [A: complex,B: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(A),insert_complex(B,A4))
    <=> ( ( A = B )
        | aa_set_complex_o(member_complex(A),A4) ) ) ).

% insert_iff
tff(fact_163_insert__iff,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A4))
    <=> ( ( A = B )
        | aa_set_real_o(member_real(A),A4) ) ) ).

% insert_iff
tff(fact_164_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A4))
    <=> ( ( (A)
        <=> (B) )
        | aa_set_o_o(member_o((A)),A4) ) ) ).

% insert_iff
tff(fact_165_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A4))
    <=> ( ( A = B )
        | aa_set_nat_o(member_nat(A),A4) ) ) ).

% insert_iff
tff(fact_166_insert__iff,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A4))
    <=> ( ( A = B )
        | aa_set_int_o(member_int(A),A4) ) ) ).

% insert_iff
tff(fact_167_insertCI,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
       => ( A = B ) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B4)) ) ).

% insertCI
tff(fact_168_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
       => ( A = B ) )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B4)) ) ).

% insertCI
tff(fact_169_insertCI,axiom,
    ! [A: complex,B4: set_complex,B: complex] :
      ( ( ~ aa_set_complex_o(member_complex(A),B4)
       => ( A = B ) )
     => aa_set_complex_o(member_complex(A),insert_complex(B,B4)) ) ).

% insertCI
tff(fact_170_insertCI,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( ~ aa_set_real_o(member_real(A),B4)
       => ( A = B ) )
     => aa_set_real_o(member_real(A),insert_real(B,B4)) ) ).

% insertCI
tff(fact_171_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ aa_set_o_o(member_o((A)),B4)
       => ( (A)
        <=> (B) ) )
     => aa_set_o_o(member_o((A)),insert_o((B),B4)) ) ).

% insertCI
tff(fact_172_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B4)
       => ( A = B ) )
     => aa_set_nat_o(member_nat(A),insert_nat(B,B4)) ) ).

% insertCI
tff(fact_173_insertCI,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A),B4)
       => ( A = B ) )
     => aa_set_int_o(member_int(A),insert_int(B,B4)) ) ).

% insertCI
tff(fact_174_case4_I5_J,axiom,
    m = aa_nat_nat(suc,na) ).

% case4(5)
tff(fact_175_case4_I6_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% case4(6)
tff(fact_176_ex__in__conv,axiom,
    ! [A4: set_complex] :
      ( ? [X4: complex] : aa_set_complex_o(member_complex(X4),A4)
    <=> ( A4 != bot_bot_set_complex ) ) ).

% ex_in_conv
tff(fact_177_ex__in__conv,axiom,
    ! [A4: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),A4)
    <=> ( A4 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_178_ex__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ? [X4: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
    <=> ( A4 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_179_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ? [X4: $o] : aa_set_o_o(member_o((X4)),A4)
    <=> ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_180_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A4)
    <=> ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_181_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A4)
    <=> ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_182_equals0I,axiom,
    ! [A4: set_complex] :
      ( ! [Y3: complex] : ~ aa_set_complex_o(member_complex(Y3),A4)
     => ( A4 = bot_bot_set_complex ) ) ).

% equals0I
tff(fact_183_equals0I,axiom,
    ! [A4: set_real] :
      ( ! [Y3: real] : ~ aa_set_real_o(member_real(Y3),A4)
     => ( A4 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_184_equals0I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A4)
     => ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_185_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y3: $o] : ~ aa_set_o_o(member_o((Y3)),A4)
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_186_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A4)
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_187_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A4)
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_188_equals0D,axiom,
    ! [A4: set_complex,A: complex] :
      ( ( A4 = bot_bot_set_complex )
     => ~ aa_set_complex_o(member_complex(A),A4) ) ).

% equals0D
tff(fact_189_equals0D,axiom,
    ! [A4: set_real,A: real] :
      ( ( A4 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A),A4) ) ).

% equals0D
tff(fact_190_equals0D,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4) ) ).

% equals0D
tff(fact_191_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A4) ) ).

% equals0D
tff(fact_192_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A4) ) ).

% equals0D
tff(fact_193_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A4) ) ).

% equals0D
tff(fact_194_emptyE,axiom,
    ! [A: complex] : ~ aa_set_complex_o(member_complex(A),bot_bot_set_complex) ).

% emptyE
tff(fact_195_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real(A),bot_bot_set_real) ).

% emptyE
tff(fact_196_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat) ).

% emptyE
tff(fact_197_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_198_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_199_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_200_Collect__mono__iff,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q))
    <=> ! [X4: complex] :
          ( aa_complex_o(P,X4)
         => aa_complex_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_201_Collect__mono__iff,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q))
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P,X4)
         => aa_Pro6150637604445964947_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_202_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
         => aa_set_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_203_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))
    <=> ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
         => aa_list_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_204_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))
    <=> ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_205_set__eq__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
        & aa_set_nat_o(ord_less_eq_set_nat(B4),A4) ) ) ).

% set_eq_subset
tff(fact_206_subset__trans,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C2)
       => aa_set_nat_o(ord_less_eq_set_nat(A4),C2) ) ) ).

% subset_trans
tff(fact_207_Collect__mono,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X5: complex] :
          ( aa_complex_o(P,X5)
         => aa_complex_o(Q,X5) )
     => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q)) ) ).

% Collect_mono
tff(fact_208_Collect__mono,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ! [X5: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P,X5)
         => aa_Pro6150637604445964947_nat_o(Q,X5) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q)) ) ).

% Collect_mono
tff(fact_209_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X5: set_nat] :
          ( aa_set_nat_o(P,X5)
         => aa_set_nat_o(Q,X5) )
     => ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_210_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X5: list_nat] :
          ( aa_list_nat_o(P,X5)
         => aa_list_nat_o(Q,X5) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_211_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_nat_o(P,X5)
         => aa_nat_o(Q,X5) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_212_subset__refl,axiom,
    ! [A4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A4),A4) ).

% subset_refl
tff(fact_213_subset__iff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
    <=> ! [T: complex] :
          ( aa_set_complex_o(member_complex(T),A4)
         => aa_set_complex_o(member_complex(T),B4) ) ) ).

% subset_iff
tff(fact_214_subset__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real(T),A4)
         => aa_set_real_o(member_real(T),B4) ) ) ).

% subset_iff
tff(fact_215_subset__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
    <=> ! [T: $o] :
          ( aa_set_o_o(member_o((T)),A4)
         => aa_set_o_o(member_o((T)),B4) ) ) ).

% subset_iff
tff(fact_216_subset__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A4)
         => aa_set_int_o(member_int(T),B4) ) ) ).

% subset_iff
tff(fact_217_subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A4)
         => aa_set_nat_o(member_nat(T),B4) ) ) ).

% subset_iff
tff(fact_218_equalityD2,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
     => aa_set_nat_o(ord_less_eq_set_nat(B4),A4) ) ).

% equalityD2
tff(fact_219_equalityD1,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
     => aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ).

% equalityD1
tff(fact_220_subset__eq,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),A4)
         => aa_set_complex_o(member_complex(X4),B4) ) ) ).

% subset_eq
tff(fact_221_subset__eq,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A4)
         => aa_set_real_o(member_real(X4),B4) ) ) ).

% subset_eq
tff(fact_222_subset__eq,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A4)
         => aa_set_o_o(member_o((X4)),B4) ) ) ).

% subset_eq
tff(fact_223_subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_eq
tff(fact_224_subset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_eq
tff(fact_225_equalityE,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B4),A4) ) ) ).

% equalityE
tff(fact_226_subsetD,axiom,
    ! [A4: set_complex,B4: set_complex,C: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(member_complex(C),A4)
       => aa_set_complex_o(member_complex(C),B4) ) ) ).

% subsetD
tff(fact_227_subsetD,axiom,
    ! [A4: set_real,B4: set_real,C: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
     => ( aa_set_real_o(member_real(C),A4)
       => aa_set_real_o(member_real(C),B4) ) ) ).

% subsetD
tff(fact_228_subsetD,axiom,
    ! [A4: set_o,B4: set_o,C: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( aa_set_o_o(member_o((C)),A4)
       => aa_set_o_o(member_o((C)),B4) ) ) ).

% subsetD
tff(fact_229_subsetD,axiom,
    ! [A4: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(member_int(C),A4)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% subsetD
tff(fact_230_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(member_nat(C),A4)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% subsetD
tff(fact_231_in__mono,axiom,
    ! [A4: set_complex,B4: set_complex,X3: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => aa_set_complex_o(member_complex(X3),B4) ) ) ).

% in_mono
tff(fact_232_in__mono,axiom,
    ! [A4: set_real,B4: set_real,X3: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
     => ( aa_set_real_o(member_real(X3),A4)
       => aa_set_real_o(member_real(X3),B4) ) ) ).

% in_mono
tff(fact_233_in__mono,axiom,
    ! [A4: set_o,B4: set_o,X3: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( aa_set_o_o(member_o((X3)),A4)
       => aa_set_o_o(member_o((X3)),B4) ) ) ).

% in_mono
tff(fact_234_in__mono,axiom,
    ! [A4: set_int,B4: set_int,X3: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(member_int(X3),A4)
       => aa_set_int_o(member_int(X3),B4) ) ) ).

% in_mono
tff(fact_235_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(member_nat(X3),A4)
       => aa_set_nat_o(member_nat(X3),B4) ) ) ).

% in_mono
tff(fact_236_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ? [B5: set_Pr1261947904930325089at_nat] :
          ( ( A4 = insert8211810215607154385at_nat(A,B5) )
          & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_237_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ? [B5: set_Pr4329608150637261639at_nat] :
          ( ( A4 = insert9069300056098147895at_nat(A,B5) )
          & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_238_mk__disjoint__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(A),A4)
     => ? [B5: set_complex] :
          ( ( A4 = insert_complex(A,B5) )
          & ~ aa_set_complex_o(member_complex(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_239_mk__disjoint__insert,axiom,
    ! [A: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ? [B5: set_real] :
          ( ( A4 = insert_real(A,B5) )
          & ~ aa_set_real_o(member_real(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_240_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ? [B5: set_o] :
          ( ( A4 = insert_o((A),B5) )
          & ~ aa_set_o_o(member_o((A)),B5) ) ) ).

% mk_disjoint_insert
tff(fact_241_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ? [B5: set_nat] :
          ( ( A4 = insert_nat(A,B5) )
          & ~ aa_set_nat_o(member_nat(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_242_mk__disjoint__insert,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ? [B5: set_int] :
          ( ( A4 = insert_int(A,B5) )
          & ~ aa_set_int_o(member_int(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_243_insert__commute,axiom,
    ! [X3: nat,Y: nat,A4: set_nat] : insert_nat(X3,insert_nat(Y,A4)) = insert_nat(Y,insert_nat(X3,A4)) ).

% insert_commute
tff(fact_244_insert__commute,axiom,
    ! [X3: int,Y: int,A4: set_int] : insert_int(X3,insert_int(Y,A4)) = insert_int(Y,insert_int(X3,A4)) ).

% insert_commute
tff(fact_245_insert__commute,axiom,
    ! [X3: $o,Y: $o,A4: set_o] : insert_o((X3),insert_o((Y),A4)) = insert_o((Y),insert_o((X3),A4)) ).

% insert_commute
tff(fact_246_insert__commute,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X3,insert8211810215607154385at_nat(Y,A4)) = insert8211810215607154385at_nat(Y,insert8211810215607154385at_nat(X3,A4)) ).

% insert_commute
tff(fact_247_insert__commute,axiom,
    ! [X3: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X3,insert9069300056098147895at_nat(Y,A4)) = insert9069300056098147895at_nat(Y,insert9069300056098147895at_nat(X3,A4)) ).

% insert_commute
tff(fact_248_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),B4)
       => ( ( insert8211810215607154385at_nat(A,A4) = insert8211810215607154385at_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C3: set_Pr1261947904930325089at_nat] :
                ( ( A4 = insert8211810215607154385at_nat(B,C3) )
                & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),C3)
                & ( B4 = insert8211810215607154385at_nat(A,C3) )
                & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_249_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),B4)
       => ( ( insert9069300056098147895at_nat(A,A4) = insert9069300056098147895at_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C3: set_Pr4329608150637261639at_nat] :
                ( ( A4 = insert9069300056098147895at_nat(B,C3) )
                & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),C3)
                & ( B4 = insert9069300056098147895at_nat(A,C3) )
                & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_250_insert__eq__iff,axiom,
    ! [A: complex,A4: set_complex,B: complex,B4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A),A4)
     => ( ~ aa_set_complex_o(member_complex(B),B4)
       => ( ( insert_complex(A,A4) = insert_complex(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C3: set_complex] :
                ( ( A4 = insert_complex(B,C3) )
                & ~ aa_set_complex_o(member_complex(B),C3)
                & ( B4 = insert_complex(A,C3) )
                & ~ aa_set_complex_o(member_complex(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_251_insert__eq__iff,axiom,
    ! [A: real,A4: set_real,B: real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(A),A4)
     => ( ~ aa_set_real_o(member_real(B),B4)
       => ( ( insert_real(A,A4) = insert_real(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C3: set_real] :
                ( ( A4 = insert_real(B,C3) )
                & ~ aa_set_real_o(member_real(B),C3)
                & ( B4 = insert_real(A,C3) )
                & ~ aa_set_real_o(member_real(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_252_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A4)
     => ( ~ aa_set_o_o(member_o((B)),B4)
       => ( ( insert_o((A),A4) = insert_o((B),B4) )
        <=> $ite(
              (A) = (B),
              A4 = B4,
              ? [C3: set_o] :
                ( ( A4 = insert_o((B),C3) )
                & ~ aa_set_o_o(member_o((B)),C3)
                & ( B4 = insert_o((A),C3) )
                & ~ aa_set_o_o(member_o((A)),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_253_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A4)
     => ( ~ aa_set_nat_o(member_nat(B),B4)
       => ( ( insert_nat(A,A4) = insert_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C3: set_nat] :
                ( ( A4 = insert_nat(B,C3) )
                & ~ aa_set_nat_o(member_nat(B),C3)
                & ( B4 = insert_nat(A,C3) )
                & ~ aa_set_nat_o(member_nat(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_254_insert__eq__iff,axiom,
    ! [A: int,A4: set_int,B: int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),A4)
     => ( ~ aa_set_int_o(member_int(B),B4)
       => ( ( insert_int(A,A4) = insert_int(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C3: set_int] :
                ( ( A4 = insert_int(B,C3) )
                & ~ aa_set_int_o(member_int(B),C3)
                & ( B4 = insert_int(A,C3) )
                & ~ aa_set_int_o(member_int(A),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_255_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( insert8211810215607154385at_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_256_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( insert9069300056098147895at_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_257_insert__absorb,axiom,
    ! [A: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(A),A4)
     => ( insert_complex(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_258_insert__absorb,axiom,
    ! [A: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ( insert_real(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_259_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( insert_o((A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_260_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( insert_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_261_insert__absorb,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( insert_int(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_262_insert__ident,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B4)
       => ( ( insert8211810215607154385at_nat(X3,A4) = insert8211810215607154385at_nat(X3,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_263_insert__ident,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),B4)
       => ( ( insert9069300056098147895at_nat(X3,A4) = insert9069300056098147895at_nat(X3,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_264_insert__ident,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(X3),A4)
     => ( ~ aa_set_complex_o(member_complex(X3),B4)
       => ( ( insert_complex(X3,A4) = insert_complex(X3,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_265_insert__ident,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(X3),A4)
     => ( ~ aa_set_real_o(member_real(X3),B4)
       => ( ( insert_real(X3,A4) = insert_real(X3,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_266_insert__ident,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X3)),A4)
     => ( ~ aa_set_o_o(member_o((X3)),B4)
       => ( ( insert_o((X3),A4) = insert_o((X3),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_267_insert__ident,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X3),A4)
     => ( ~ aa_set_nat_o(member_nat(X3),B4)
       => ( ( insert_nat(X3,A4) = insert_nat(X3,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_268_insert__ident,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X3),A4)
     => ( ~ aa_set_int_o(member_int(X3),B4)
       => ( ( insert_int(X3,A4) = insert_int(X3,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_269_Set_Oset__insert,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( A4 = insert8211810215607154385at_nat(X3,B5) )
           => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B5) ) ) ).

% Set.set_insert
tff(fact_270_Set_Oset__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4)
     => ~ ! [B5: set_Pr4329608150637261639at_nat] :
            ( ( A4 = insert9069300056098147895at_nat(X3,B5) )
           => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),B5) ) ) ).

% Set.set_insert
tff(fact_271_Set_Oset__insert,axiom,
    ! [X3: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(X3),A4)
     => ~ ! [B5: set_complex] :
            ( ( A4 = insert_complex(X3,B5) )
           => aa_set_complex_o(member_complex(X3),B5) ) ) ).

% Set.set_insert
tff(fact_272_Set_Oset__insert,axiom,
    ! [X3: real,A4: set_real] :
      ( aa_set_real_o(member_real(X3),A4)
     => ~ ! [B5: set_real] :
            ( ( A4 = insert_real(X3,B5) )
           => aa_set_real_o(member_real(X3),B5) ) ) ).

% Set.set_insert
tff(fact_273_Set_Oset__insert,axiom,
    ! [X3: $o,A4: set_o] :
      ( aa_set_o_o(member_o((X3)),A4)
     => ~ ! [B5: set_o] :
            ( ( A4 = insert_o((X3),B5) )
           => aa_set_o_o(member_o((X3)),B5) ) ) ).

% Set.set_insert
tff(fact_274_Set_Oset__insert,axiom,
    ! [X3: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(X3),A4)
     => ~ ! [B5: set_nat] :
            ( ( A4 = insert_nat(X3,B5) )
           => aa_set_nat_o(member_nat(X3),B5) ) ) ).

% Set.set_insert
tff(fact_275_Set_Oset__insert,axiom,
    ! [X3: int,A4: set_int] :
      ( aa_set_int_o(member_int(X3),A4)
     => ~ ! [B5: set_int] :
            ( ( A4 = insert_int(X3,B5) )
           => aa_set_int_o(member_int(X3),B5) ) ) ).

% Set.set_insert
tff(fact_276_insertI2,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B4)) ) ).

% insertI2
tff(fact_277_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B4)) ) ).

% insertI2
tff(fact_278_insertI2,axiom,
    ! [A: complex,B4: set_complex,B: complex] :
      ( aa_set_complex_o(member_complex(A),B4)
     => aa_set_complex_o(member_complex(A),insert_complex(B,B4)) ) ).

% insertI2
tff(fact_279_insertI2,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( aa_set_real_o(member_real(A),B4)
     => aa_set_real_o(member_real(A),insert_real(B,B4)) ) ).

% insertI2
tff(fact_280_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( aa_set_o_o(member_o((A)),B4)
     => aa_set_o_o(member_o((A)),insert_o((B),B4)) ) ).

% insertI2
tff(fact_281_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A),B4)
     => aa_set_nat_o(member_nat(A),insert_nat(B,B4)) ) ).

% insertI2
tff(fact_282_insertI2,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( aa_set_int_o(member_int(A),B4)
     => aa_set_int_o(member_int(A),insert_int(B,B4)) ) ).

% insertI2
tff(fact_283_insertI1,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,B4)) ).

% insertI1
tff(fact_284_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,B4)) ).

% insertI1
tff(fact_285_insertI1,axiom,
    ! [A: complex,B4: set_complex] : aa_set_complex_o(member_complex(A),insert_complex(A,B4)) ).

% insertI1
tff(fact_286_insertI1,axiom,
    ! [A: real,B4: set_real] : aa_set_real_o(member_real(A),insert_real(A,B4)) ).

% insertI1
tff(fact_287_insertI1,axiom,
    ! [A: $o,B4: set_o] : aa_set_o_o(member_o((A)),insert_o((A),B4)) ).

% insertI1
tff(fact_288_insertI1,axiom,
    ! [A: nat,B4: set_nat] : aa_set_nat_o(member_nat(A),insert_nat(A,B4)) ).

% insertI1
tff(fact_289_insertI1,axiom,
    ! [A: int,B4: set_int] : aa_set_int_o(member_int(A),insert_int(A,B4)) ).

% insertI1
tff(fact_290_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A4))
     => ( ( A != B )
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4) ) ) ).

% insertE
tff(fact_291_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A4))
     => ( ( A != B )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4) ) ) ).

% insertE
tff(fact_292_insertE,axiom,
    ! [A: complex,B: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(A),insert_complex(B,A4))
     => ( ( A != B )
       => aa_set_complex_o(member_complex(A),A4) ) ) ).

% insertE
tff(fact_293_insertE,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A4))
     => ( ( A != B )
       => aa_set_real_o(member_real(A),A4) ) ) ).

% insertE
tff(fact_294_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A4))
     => ( ( (A)
        <=> ~ (B) )
       => aa_set_o_o(member_o((A)),A4) ) ) ).

% insertE
tff(fact_295_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A4))
     => ( ( A != B )
       => aa_set_nat_o(member_nat(A),A4) ) ) ).

% insertE
tff(fact_296_insertE,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A4))
     => ( ( A != B )
       => aa_set_int_o(member_int(A),A4) ) ) ).

% insertE
tff(fact_297_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_298_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_299_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( insert_o((A),bot_bot_set_o) = insert_o((B),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_300_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_301_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_302_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A4) != bot_bo228742789529271731at_nat ).

% insert_not_empty
tff(fact_303_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A4) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_304_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] : insert_o((A),A4) != bot_bot_set_o ).

% insert_not_empty
tff(fact_305_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,A4) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_306_insert__not__empty,axiom,
    ! [A: int,A4: set_int] : insert_int(A,A4) != bot_bot_set_int ).

% insert_not_empty
tff(fact_307_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(C,insert9069300056098147895at_nat(D,bot_bo228742789529271731at_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_308_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C,insert8211810215607154385at_nat(D,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_309_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( insert_o((A),insert_o((B),bot_bot_set_o)) = insert_o((C),insert_o((D),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C) )
          & ( (B)
          <=> (D) ) )
        | ( ( (A)
          <=> (D) )
          & ( (B)
          <=> (C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_310_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_311_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_312_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_313_singleton__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_set_complex_o(member_complex(B),insert_complex(A,bot_bot_set_complex))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_314_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),insert_real(A,bot_bot_set_real))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_315_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_316_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_317_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_318_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_319_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_320_singletonD,axiom,
    ! [B: complex,A: complex] :
      ( aa_set_complex_o(member_complex(B),insert_complex(A,bot_bot_set_complex))
     => ( B = A ) ) ).

% singletonD
tff(fact_321_singletonD,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),insert_real(A,bot_bot_set_real))
     => ( B = A ) ) ).

% singletonD
tff(fact_322_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_323_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_324_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_325_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_326_subset__insertI2,axiom,
    ! [A4: set_int,B4: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(A4),insert_int(B,B4)) ) ).

% subset_insertI2
tff(fact_327_subset__insertI2,axiom,
    ! [A4: set_o,B4: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => aa_set_o_o(ord_less_eq_set_o(A4),insert_o((B),B4)) ) ).

% subset_insertI2
tff(fact_328_subset__insertI2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_329_subset__insertI2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_330_subset__insertI2,axiom,
    ! [A4: set_nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_331_subset__insertI,axiom,
    ! [B4: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B4),insert_int(A,B4)) ).

% subset_insertI
tff(fact_332_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B4),insert_o((A),B4)) ).

% subset_insertI
tff(fact_333_subset__insertI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),insert8211810215607154385at_nat(A,B4)) ).

% subset_insertI
tff(fact_334_subset__insertI,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),insert9069300056098147895at_nat(A,B4)) ).

% subset_insertI
tff(fact_335_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : aa_set_nat_o(ord_less_eq_set_nat(B4),insert_nat(A,B4)) ).

% subset_insertI
tff(fact_336_subset__insert,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X3,B4))
      <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ) ).

% subset_insert
tff(fact_337_subset__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X3,B4))
      <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4) ) ) ).

% subset_insert
tff(fact_338_subset__insert,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(X3),A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),insert_complex(X3,B4))
      <=> aa_set_complex_o(ord_le211207098394363844omplex(A4),B4) ) ) ).

% subset_insert
tff(fact_339_subset__insert,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(X3),A4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),insert_real(X3,B4))
      <=> aa_set_real_o(ord_less_eq_set_real(A4),B4) ) ) ).

% subset_insert
tff(fact_340_subset__insert,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X3)),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X3),B4))
      <=> aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ) ).

% subset_insert
tff(fact_341_subset__insert,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X3),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),insert_int(X3,B4))
      <=> aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ) ).

% subset_insert
tff(fact_342_subset__insert,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X3),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(X3,B4))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ) ).

% subset_insert
tff(fact_343_insert__mono,axiom,
    ! [C2: set_int,D2: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C2),D2)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A,C2)),insert_int(A,D2)) ) ).

% insert_mono
tff(fact_344_insert__mono,axiom,
    ! [C2: set_o,D2: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C2),D2)
     => aa_set_o_o(ord_less_eq_set_o(insert_o((A),C2)),insert_o((A),D2)) ) ).

% insert_mono
tff(fact_345_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),D2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(A,C2)),insert8211810215607154385at_nat(A,D2)) ) ).

% insert_mono
tff(fact_346_insert__mono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),D2)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(A,C2)),insert9069300056098147895at_nat(A,D2)) ) ).

% insert_mono
tff(fact_347_insert__mono,axiom,
    ! [C2: set_nat,D2: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),D2)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(A,C2)),insert_nat(A,D2)) ) ).

% insert_mono
tff(fact_348_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,X3: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),X3) ).

% buildup_nothing_in_leaf
tff(fact_349_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),X3)
       => ( vEBT_vebt_maxt(Ta) = some_nat(X3) ) ) ) ).

% maxt_sound
tff(fact_350_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(X3) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),X3) ) ) ).

% maxt_corr
tff(fact_351_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),X3)
       => ( vEBT_vebt_mint(Ta) = some_nat(X3) ) ) ) ).

% mint_sound
tff(fact_352_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(X3) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),X3) ) ) ).

% mint_corr
tff(fact_353_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),X3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_354_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),X3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),X3) ) ) ) ).

% mint_corr_help
tff(fact_355_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_356_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_357_the__elem__eq,axiom,
    ! [X3: produc3843707927480180839at_nat] : the_el221668144340439132at_nat(insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)) = X3 ).

% the_elem_eq
tff(fact_358_the__elem__eq,axiom,
    ! [X3: product_prod_nat_nat] : the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)) = X3 ).

% the_elem_eq
tff(fact_359_the__elem__eq,axiom,
    ! [X3: $o] :
      ( aa_set_o_o(the_elem_o,insert_o((X3),bot_bot_set_o))
    <=> (X3) ) ).

% the_elem_eq
tff(fact_360_the__elem__eq,axiom,
    ! [X3: nat] : aa_set_nat_nat(the_elem_nat,insert_nat(X3,bot_bot_set_nat)) = X3 ).

% the_elem_eq
tff(fact_361_the__elem__eq,axiom,
    ! [X3: int] : the_elem_int(insert_int(X3,bot_bot_set_int)) = X3 ).

% the_elem_eq
tff(fact_362_dual__order_Orefl,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% dual_order.refl
tff(fact_363_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_364_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_365_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_366_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_367_even__odd__cases,axiom,
    ! [X3: nat] :
      ( ! [N: nat] : X3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : X3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_368_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_369_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(X3) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X3) ) ).

% maxbmo
tff(fact_370_ac,axiom,
    ! [Ta: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),H)
     => ( aa_nat_o(vEBT_invar_vebt(K2),H)
       => ( ( vEBT_VEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(K2) )
         => ( vEBT_vebt_mint(Ta) = vEBT_vebt_mint(K2) ) ) ) ) ).

% ac
tff(fact_371_ad,axiom,
    ! [Ta: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),H)
     => ( aa_nat_o(vEBT_invar_vebt(K2),H)
       => ( ( vEBT_VEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(K2) )
         => ( vEBT_vebt_maxt(Ta) = vEBT_vebt_maxt(K2) ) ) ) ) ).

% ad
tff(fact_372_order__refl,axiom,
    ! [X3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(X3),X3) ).

% order_refl
tff(fact_373_order__refl,axiom,
    ! [X3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),X3) ).

% order_refl
tff(fact_374_order__refl,axiom,
    ! [X3: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),X3) ).

% order_refl
tff(fact_375_order__refl,axiom,
    ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),X3) ).

% order_refl
tff(fact_376_order__refl,axiom,
    ! [X3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),X3) ).

% order_refl
tff(fact_377_bot__set__def,axiom,
    bot_bot_set_complex = collect_complex(bot_bo8750099748576881071plex_o) ).

% bot_set_def
tff(fact_378_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_379_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_380_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).

% bot_set_def
tff(fact_381_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_382_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_383_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_384_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_385_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_386_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_387_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_388_le__cases3,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X3) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X3)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_389_le__cases3,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),X3) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),X3)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_390_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X3) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X3)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_391_le__cases3,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X3) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X3)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_392_order__class_Oorder__eq__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( X3 = Y )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
        & aa_set_nat_o(ord_less_eq_set_nat(Y),X3) ) ) ).

% order_class.order_eq_iff
tff(fact_393_order__class_Oorder__eq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) ) ) ).

% order_class.order_eq_iff
tff(fact_394_order__class_Oorder__eq__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) ) ) ).

% order_class.order_eq_iff
tff(fact_395_order__class_Oorder__eq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ) ).

% order_class.order_eq_iff
tff(fact_396_order__class_Oorder__eq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) ) ) ).

% order_class.order_eq_iff
tff(fact_397_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% ord_eq_le_trans
tff(fact_398_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_399_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_400_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_401_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_402_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( B = C )
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% ord_le_eq_trans
tff(fact_403_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_404_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_405_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_406_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_407_order__antisym,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_408_order__antisym,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_409_order__antisym,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3)
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_410_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_411_order__antisym,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_412_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% order.trans
tff(fact_413_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_414_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% order.trans
tff(fact_415_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_416_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_417_order__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_eq_set_nat(X3),Z) ) ) ).

% order_trans
tff(fact_418_order__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) ) ) ).

% order_trans
tff(fact_419_order__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Z) ) ) ).

% order_trans
tff(fact_420_order__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z) ) ) ).

% order_trans
tff(fact_421_order__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z) ) ) ).

% order_trans
tff(fact_422_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
     => ( ! [A3: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_423_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
     => ( ! [A3: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B3),A3)
           => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
       => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_424_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
     => ( ! [A3: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_425_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
     => ( ! [A3: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B3),A3)
           => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
       => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_426_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.eq_iff
tff(fact_427_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_428_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_429_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_430_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_431_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_432_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_433_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_434_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_435_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_436_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% dual_order.trans
tff(fact_437_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_438_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% dual_order.trans
tff(fact_439_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_440_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_441_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_442_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_443_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_444_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_445_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_446_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_447_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_448_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_449_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_450_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_451_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_452_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_453_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_454_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_455_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_456_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_457_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_458_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X5)),aa_int_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_459_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_460_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_461_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_462_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_463_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_464_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_465_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_466_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_467_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_468_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_469_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_470_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_471_order__eq__refl,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( X3 = Y )
     => aa_set_nat_o(ord_less_eq_set_nat(X3),Y) ) ).

% order_eq_refl
tff(fact_472_order__eq__refl,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ).

% order_eq_refl
tff(fact_473_order__eq__refl,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y) ) ).

% order_eq_refl
tff(fact_474_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ).

% order_eq_refl
tff(fact_475_order__eq__refl,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ).

% order_eq_refl
tff(fact_476_linorder__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) ) ).

% linorder_linear
tff(fact_477_linorder__linear,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) ) ).

% linorder_linear
tff(fact_478_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ).

% linorder_linear
tff(fact_479_linorder__linear,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) ) ).

% linorder_linear
tff(fact_480_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_481_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_482_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_483_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_484_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_485_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_486_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_487_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_488_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_489_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_490_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_491_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_492_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_493_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_494_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_495_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_496_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_497_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_498_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_499_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_500_linorder__le__cases,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) ) ).

% linorder_le_cases
tff(fact_501_linorder__le__cases,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) ) ).

% linorder_le_cases
tff(fact_502_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ).

% linorder_le_cases
tff(fact_503_linorder__le__cases,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) ) ).

% linorder_le_cases
tff(fact_504_order__antisym__conv,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_505_order__antisym__conv,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_506_order__antisym__conv,axiom,
    ! [Y: num,X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_507_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_508_order__antisym__conv,axiom,
    ! [Y: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_509_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A) ).

% bot.extremum
tff(fact_510_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_511_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_512_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_513_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_514_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_515_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_516_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_517_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_518_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_519_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_520_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_521_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_522_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_523_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_524_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),X3)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),X3)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),X3) ) ) ) ).

% member_valid_both_member_options
tff(fact_525_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Maa) ).

% vebt_maxt.simps(3)
tff(fact_526_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Mia) ).

% vebt_mint.simps(3)
tff(fact_527_nat__add__left__cancel__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_528_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_529_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_530_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_531_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_532_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_533_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_534_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_535_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_536_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_537_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_538_add__shift,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X3)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_539_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X3)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),X3)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),X3) ) ) ).

% both_member_options_def
tff(fact_540_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_541_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_542_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,X3: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),X3) ).

% buildup_nothing_in_min_max
tff(fact_543_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_544_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_545_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_546_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_547_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_548_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_549_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_550_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_551_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_552_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_553_not__None__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( X3 != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : X3 = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_554_not__None__eq,axiom,
    ! [X3: option_nat] :
      ( ( X3 != none_nat )
    <=> ? [Y4: nat] : X3 = some_nat(Y4) ) ).

% not_None_eq
tff(fact_555_not__None__eq,axiom,
    ! [X3: option_num] :
      ( ( X3 != none_num )
    <=> ? [Y4: num] : X3 = some_num(Y4) ) ).

% not_None_eq
tff(fact_556_not__Some__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : X3 != some_P7363390416028606310at_nat(Y4)
    <=> ( X3 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_557_not__Some__eq,axiom,
    ! [X3: option_nat] :
      ( ! [Y4: nat] : X3 != some_nat(Y4)
    <=> ( X3 = none_nat ) ) ).

% not_Some_eq
tff(fact_558_not__Some__eq,axiom,
    ! [X3: option_num] :
      ( ! [Y4: num] : X3 != some_num(Y4)
    <=> ( X3 = none_num ) ) ).

% not_Some_eq
tff(fact_559_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_560_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc3739950874144486240at_nat] :
      ( ! [Uu: fun_Pr2028404973800837934_nat_o,Uv: option4927543243414619207at_nat] : X3 != produc3653971535690468304at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2028404973800837934_nat_o,V: product_prod_nat_nat] : X3 != produc3653971535690468304at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X5: product_prod_nat_nat,Y3: product_prod_nat_nat] : X3 != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X5),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_561_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc1615844007061411358on_nat] :
      ( ! [Uu: fun_nat_fun_nat_o,Uv: option_nat] : X3 != produc7967433646104525454on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_o,V: nat] : X3 != produc7967433646104525454on_nat(Uw,produc5098337634421038937on_nat(some_nat(V),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X5: nat,Y3: nat] : X3 != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X5),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_562_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc50614890452615238on_num] :
      ( ! [Uu: fun_num_fun_num_o,Uv: option_num] : X3 != produc1140783415258738358on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_o,V: num] : X3 != produc1140783415258738358on_num(Uw,produc8585076106096196333on_num(some_num(V),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X5: num,Y3: num] : X3 != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X5),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_563_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc1260151622200656695at_nat] :
      ( ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : X3 != produc7814092408141220007at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : X3 != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] : X3 != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_564_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc7004362771917097900on_nat] :
      ( ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : X3 != produc6699827698519632094on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_nat,V: nat] : X3 != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(some_nat(V),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat] : X3 != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_565_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc2703417854250991842on_num] :
      ( ! [Uu: fun_num_fun_num_num,Uv: option_num] : X3 != produc6360833703471310868on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_num,V: num] : X3 != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(some_num(V),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B3: num] : X3 != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A3),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_566_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_567_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_568_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_569_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_570_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_571_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_572_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_573_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] : Y != some_nat(X22) ) ).

% option.exhaust
tff(fact_574_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != some_num(X22) ) ).

% option.exhaust
tff(fact_575_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_ex
tff(fact_576_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_ex
tff(fact_577_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_578_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_all
tff(fact_579_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_all
tff(fact_580_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_all
tff(fact_581_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X3),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X3),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X3 = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X3),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_582_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X3),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X3),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X3 = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X3),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_583_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X3),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X3),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X3 = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X3),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_584_combine__options__cases,axiom,
    ! [X3: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X3),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X3),Y) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X3 = some_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X3),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_585_combine__options__cases,axiom,
    ! [X3: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( X3 = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X3),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X3),Y) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X3 = some_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X3),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_586_combine__options__cases,axiom,
    ! [X3: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( X3 = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X3),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X3),Y) )
       => ( ! [A3: nat,B3: num] :
              ( ( X3 = some_nat(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X3),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_587_combine__options__cases,axiom,
    ! [X3: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X3),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X3),Y) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X3 = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X3),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_588_combine__options__cases,axiom,
    ! [X3: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( X3 = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X3),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X3),Y) )
       => ( ! [A3: num,B3: nat] :
              ( ( X3 = some_num(A3) )
             => ( ( Y = some_nat(B3) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X3),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_589_combine__options__cases,axiom,
    ! [X3: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( X3 = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X3),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X3),Y) )
       => ( ! [A3: num,B3: num] :
              ( ( X3 = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X3),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X3),Y) ) ) ) ).

% combine_options_cases
tff(fact_590_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_591_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_592_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_593_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_594_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_595_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_596_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_597_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_598_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(plus_plus_real(B),aa_real_real(plus_plus_real(A),C)) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.left_commute
tff(fact_599_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.left_commute
tff(fact_600_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_601_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_602_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(B),A) ).

% add.commute
tff(fact_603_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% add.commute
tff(fact_604_add_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% add.commute
tff(fact_605_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% add.commute
tff(fact_606_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_607_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_608_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_609_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_610_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_611_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_612_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.assoc
tff(fact_613_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_614_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_615_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_616_group__cancel_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(plus_plus_real(K2),B) )
     => ( aa_real_real(plus_plus_real(A),B4) = aa_real_real(plus_plus_real(K2),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add2
tff(fact_617_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_618_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_619_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_620_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(plus_plus_real(A4),B) = aa_real_real(plus_plus_real(K2),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add1
tff(fact_621_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_622_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_623_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_624_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_real_real(plus_plus_real(I),K2) = aa_real_real(plus_plus_real(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_625_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_626_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_627_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_628_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_629_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_630_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_631_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_632_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_633_Suc__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X3) = aa_nat_nat(suc,Y) )
     => ( X3 = Y ) ) ).

% Suc_inject
tff(fact_634_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X5: nat] :
            ( aa_nat_o(P,X5)
            & ! [Y5: nat] :
                ( aa_nat_o(P,Y5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y5),X5) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_635_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_636_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_637_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_638_le__trans,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2) ) ) ).

% le_trans
tff(fact_639_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_640_size__neq__size__imp__neq,axiom,
    ! [X3: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X3) != size_s6755466524823107622T_VEBT(Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_641_size__neq__size__imp__neq,axiom,
    ! [X3: list_o,Y: list_o] :
      ( ( size_size_list_o(X3) != size_size_list_o(Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_642_size__neq__size__imp__neq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X3) != size_size_list_nat(Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_643_size__neq__size__imp__neq,axiom,
    ! [X3: list_int,Y: list_int] :
      ( ( size_size_list_int(X3) != size_size_list_int(Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_644_size__neq__size__imp__neq,axiom,
    ! [X3: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X3) != aa_num_nat(size_size_num,Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_645_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_646_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_647_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_648_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_649_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_650_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_651_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_652_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_653_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_654_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_right_mono
tff(fact_655_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_656_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_657_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_658_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C5: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C5) ) ).

% less_eqE
tff(fact_659_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_left_mono
tff(fact_660_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_661_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_662_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_663_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_mono
tff(fact_664_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_mono
tff(fact_665_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_666_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_667_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_668_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),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),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_669_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_670_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K2: 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,K2),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),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_671_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_672_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),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),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_673_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_674_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),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),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_675_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_676_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K2 = 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),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_677_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K2 = 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_678_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K2 = 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),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_679_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X5: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X5),X5)
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X5),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X5),Z2) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_680_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_681_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_682_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_683_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_684_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_685_Suc__le__D,axiom,
    ! [Nb: nat,M3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M3)
     => ? [M: nat] : M3 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_686_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_687_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_688_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_689_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_690_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_691_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_692_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_693_trans__le__add2,axiom,
    ! [I: nat,J: nat,Mb: 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,Mb),J)) ) ).

% trans_le_add2
tff(fact_694_trans__le__add1,axiom,
    ! [I: nat,J: nat,Mb: 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),Mb)) ) ).

% trans_le_add1
tff(fact_695_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_le_mono1
tff(fact_696_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: 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,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_697_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N) ) ).

% le_Suc_ex
tff(fact_698_add__leD2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% add_leD2
tff(fact_699_add__leD1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_700_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_701_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_702_add__leE,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% add_leE
tff(fact_703_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_704_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_705_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_706_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_707_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_708_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_709_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_710_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_711_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_712_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_713_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Maa),X3)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_714_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_715_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_716_Collect__empty__eq__bot,axiom,
    ! [P: fun_complex_o] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ( P = bot_bo8750099748576881071plex_o ) ) ).

% Collect_empty_eq_bot
tff(fact_717_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_718_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_719_Collect__empty__eq__bot,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ( P = bot_bo5074786513356115630_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_720_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_721_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_722_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_723_bot__empty__eq,axiom,
    ! [X: complex] :
      ( aa_complex_o(bot_bo8750099748576881071plex_o,X)
    <=> aa_set_complex_o(member_complex(X),bot_bot_set_complex) ) ).

% bot_empty_eq
tff(fact_724_bot__empty__eq,axiom,
    ! [X: real] :
      ( aa_real_o(bot_bot_fun_real_o,X)
    <=> aa_set_real_o(member_real(X),bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_725_bot__empty__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(bot_bo5074786513356115630_nat_o,X)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq
tff(fact_726_bot__empty__eq,axiom,
    ! [X: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X))
    <=> aa_set_o_o(member_o((X)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_727_bot__empty__eq,axiom,
    ! [X: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X)
    <=> aa_set_nat_o(member_nat(X),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_728_bot__empty__eq,axiom,
    ! [X: int] :
      ( aa_int_o(bot_bot_fun_int_o,X)
    <=> aa_set_int_o(member_int(X),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_729_is__singleton__the__elem,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A4)
    <=> ( A4 = insert9069300056098147895at_nat(the_el221668144340439132at_nat(A4),bot_bo228742789529271731at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_730_is__singleton__the__elem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4)
    <=> ( A4 = insert8211810215607154385at_nat(the_el2281957884133575798at_nat(A4),bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_731_is__singleton__the__elem,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
    <=> ( A4 = insert_o(aa_set_o_o(the_elem_o,A4),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_732_is__singleton__the__elem,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ( A4 = insert_nat(aa_set_nat_nat(the_elem_nat,A4),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_733_is__singleton__the__elem,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
    <=> ( A4 = insert_int(the_elem_int(A4),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_734_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_735_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_736_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_737_sprop1,axiom,
    ( ( sa = vEBT_Node(info,deg,treeList,summary) )
    & ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),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) )
    & aa_nat_o(vEBT_invar_vebt(summary),m)
    & ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
       => aa_nat_o(vEBT_invar_vebt(X),na) ) ) ).

% sprop1
tff(fact_738_is__singletonI,axiom,
    ! [X3: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)) ).

% is_singletonI
tff(fact_739_is__singletonI,axiom,
    ! [X3: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)) ).

% is_singletonI
tff(fact_740_is__singletonI,axiom,
    ! [X3: $o] : aa_set_o_o(is_singleton_o,insert_o((X3),bot_bot_set_o)) ).

% is_singletonI
tff(fact_741_is__singletonI,axiom,
    ! [X3: nat] : aa_set_nat_o(is_singleton_nat,insert_nat(X3,bot_bot_set_nat)) ).

% is_singletonI
tff(fact_742_is__singletonI,axiom,
    ! [X3: int] : aa_set_int_o(is_singleton_int,insert_int(X3,bot_bot_set_int)) ).

% is_singletonI
tff(fact_743_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_13) ) ).

% not_min_Null_member
tff(fact_744_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,X3: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),X3) ) ).

% min_Null_member
tff(fact_745_power__shift,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X3)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_746_case4_I4_J,axiom,
    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) ).

% case4(4)
tff(fact_747_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_748_a0,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg) ).

% a0
tff(fact_749__092_060open_062length_AtreeList_H_A_061_A2_A_094_Am_092_060close_062,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% \<open>length treeList' = 2 ^ m\<close>
tff(fact_750__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Info: option4927543243414619207at_nat] :
        ~ ( ( sa = vEBT_Node(Info,deg,TreeList2,Summary) )
          & ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),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) )
          & aa_nat_o(vEBT_invar_vebt(Summary),m)
          & ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
             => aa_nat_o(vEBT_invar_vebt(X),na) ) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_751_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,X3) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),X3),Sx) ) ) ).

% succ_corr
tff(fact_752_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,X3) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Ta)),X3),Sx) ) ) ).

% succ_correct
tff(fact_753_case4_I10_J,axiom,
    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)) ).

% case4(10)
tff(fact_754_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X5),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X5: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_755_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X5),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X5: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_756_is__singletonI_H,axiom,
    ! [A4: set_complex] :
      ( ( A4 != bot_bot_set_complex )
     => ( ! [X5: complex,Y3: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => ( aa_set_complex_o(member_complex(Y3),A4)
             => ( X5 = Y3 ) ) )
       => aa_set_complex_o(is_singleton_complex,A4) ) ) ).

% is_singletonI'
tff(fact_757_is__singletonI_H,axiom,
    ! [A4: set_real] :
      ( ( A4 != bot_bot_set_real )
     => ( ! [X5: real,Y3: real] :
            ( aa_set_real_o(member_real(X5),A4)
           => ( aa_set_real_o(member_real(Y3),A4)
             => ( X5 = Y3 ) ) )
       => aa_set_real_o(is_singleton_real,A4) ) ) ).

% is_singletonI'
tff(fact_758_is__singletonI_H,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A4)
           => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A4)
             => ( X5 = Y3 ) ) )
       => aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4) ) ) ).

% is_singletonI'
tff(fact_759_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X5: $o,Y3: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
           => ( aa_set_o_o(member_o((Y3)),A4)
             => ( (X5)
              <=> (Y3) ) ) )
       => aa_set_o_o(is_singleton_o,A4) ) ) ).

% is_singletonI'
tff(fact_760_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X5: nat,Y3: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => ( aa_set_nat_o(member_nat(Y3),A4)
             => ( X5 = Y3 ) ) )
       => aa_set_nat_o(is_singleton_nat,A4) ) ) ).

% is_singletonI'
tff(fact_761_is__singletonI_H,axiom,
    ! [A4: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X5: int,Y3: int] :
            ( aa_set_int_o(member_int(X5),A4)
           => ( aa_set_int_o(member_int(Y3),A4)
             => ( X5 = Y3 ) ) )
       => aa_set_int_o(is_singleton_int,A4) ) ) ).

% is_singletonI'
tff(fact_762_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_763_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_764_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_765_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu2,none_P5556105721700978146at_nat,Uv2) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_766_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : vEBT_V819420779217536731ft_num(Uu2,none_num,Uv2) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_767_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu2),none_nat),Uv2) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_768_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,Sb: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y3)),R2)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y3)),Sb) )
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(R2),Sb) ) ).

% subrelI
tff(fact_769_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,Sb: set_Pr8218934625190621173um_num] :
      ( ! [X5: num,Y3: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X5,Y3),R2)
         => member7279096912039735102um_num(product_Pair_num_num(X5,Y3),Sb) )
     => ord_le880128212290418581um_num(R2,Sb) ) ).

% subrelI
tff(fact_770_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,Sb: set_Pr6200539531224447659at_num] :
      ( ! [X5: nat,Y3: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X5,Y3),R2)
         => member9148766508732265716at_num(product_Pair_nat_num(X5,Y3),Sb) )
     => ord_le8085105155179020875at_num(R2,Sb) ) ).

% subrelI
tff(fact_771_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Sb: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3)),R2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3)),Sb) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,R2),Sb) ) ).

% subrelI
tff(fact_772_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,Sb: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Y3),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Y3),Sb) )
     => ord_le2843351958646193337nt_int(R2,Sb) ) ).

% subrelI
tff(fact_773_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_774_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_775_is__singletonE,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A4)
     => ~ ! [X5: produc3843707927480180839at_nat] : A4 != insert9069300056098147895at_nat(X5,bot_bo228742789529271731at_nat) ) ).

% is_singletonE
tff(fact_776_is__singletonE,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4)
     => ~ ! [X5: product_prod_nat_nat] : A4 != insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat) ) ).

% is_singletonE
tff(fact_777_is__singletonE,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
     => ~ ! [X5: $o] : A4 != insert_o((X5),bot_bot_set_o) ) ).

% is_singletonE
tff(fact_778_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
     => ~ ! [X5: nat] : A4 != insert_nat(X5,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_779_is__singletonE,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
     => ~ ! [X5: int] : A4 != insert_int(X5,bot_bot_set_int) ) ).

% is_singletonE
tff(fact_780_is__singleton__def,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A4)
    <=> ? [X4: produc3843707927480180839at_nat] : A4 = insert9069300056098147895at_nat(X4,bot_bo228742789529271731at_nat) ) ).

% is_singleton_def
tff(fact_781_is__singleton__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4)
    <=> ? [X4: product_prod_nat_nat] : A4 = insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat) ) ).

% is_singleton_def
tff(fact_782_is__singleton__def,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
    <=> ? [X4: $o] : A4 = insert_o((X4),bot_bot_set_o) ) ).

% is_singleton_def
tff(fact_783_is__singleton__def,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ? [X4: nat] : A4 = insert_nat(X4,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_784_is__singleton__def,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
    <=> ? [X4: int] : A4 = insert_int(X4,bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_785_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw2,some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_786_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_num_fun_num_num,V2: num] : vEBT_V819420779217536731ft_num(Uw2,some_num(V2),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_787_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw2),some_nat(V2)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_788_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X3,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V: product_prod_nat_nat] : Xa2 = some_P7363390416028606310at_nat(V)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2 = some_P7363390416028606310at_nat(A3) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb = some_P7363390416028606310at_nat(B3) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X3,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_789_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: fun_num_fun_num_num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X3,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V: num] : Xa2 = some_num(V)
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2 = some_num(A3) )
               => ! [B3: num] :
                    ( ( Xb = some_num(B3) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X3,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_790_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: fun_nat_fun_nat_nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X3),Xa2),Xb) = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V: nat] : Xa2 = some_nat(V)
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2 = some_nat(A3) )
               => ! [B3: nat] :
                    ( ( Xb = some_nat(B3) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X3,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_791_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_792_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_793_insert__simp__mima,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( X3 = Mia )
        | ( X3 = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_794_succ__min,axiom,
    ! [Dega: nat,X3: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = some_nat(Mia) ) ) ) ).

% succ_min
tff(fact_795_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_796_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% Suc_numeral
tff(fact_797_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X5),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_798_del__single__cont,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( X3 = Mia )
        & ( X3 = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ) ) ) ).

% del_single_cont
tff(fact_799_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,X3) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpyd
tff(fact_800_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( some_nat(Mb) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_801_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X3: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),X3)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_802_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(Ta,X3)),Nb) ) ).

% delete_pres_valid
tff(fact_803_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Ta,X3)),Y)
      <=> ( ( X3 != Y )
          & aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_804_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_delete(Ta,X3)),Y)
      <=> ( ( X3 != Y )
          & aa_nat_o(vEBT_vebt_member(Ta),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_805_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_806_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Mb) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_807_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Mb) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_808_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_809_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_810_succ__member,axiom,
    ! [Ta: vEBT_VEBT,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),X3),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
        & ! [Z3: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3) ) ) ) ).

% succ_member
tff(fact_811_valid__pres__insert,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_insert(Ta,X3)),Nb) ) ) ).

% valid_pres_insert
tff(fact_812_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_813_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_814_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_815_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_816_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_817_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_818_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_819_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_820_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_821_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_822_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_823_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_824_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_825_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_826_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_827_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_828_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_829_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_830_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_831_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,X3)),X3) ) ) ).

% valid_insert_both_member_options_add
tff(fact_832_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X3)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),X3) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_833_nat__add__left__cancel__less,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_834_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,X3)),Y)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y)
              | ( X3 = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_835_delt__out__of__range,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_836_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_837_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_838_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_839_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_840_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_841_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_842_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_843_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_844_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_845_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_846_add__numeral__left,axiom,
    ! [V2: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W))),Z) ).

% add_numeral_left
tff(fact_847_add__numeral__left,axiom,
    ! [V2: num,W: num,Z: real] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V2)),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W))),Z) ).

% add_numeral_left
tff(fact_848_add__numeral__left,axiom,
    ! [V2: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W))),Z) ).

% add_numeral_left
tff(fact_849_add__numeral__left,axiom,
    ! [V2: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W))),Z) ).

% add_numeral_left
tff(fact_850_add__numeral__left,axiom,
    ! [V2: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W))),Z) ).

% add_numeral_left
tff(fact_851_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_852_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_853_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_854_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_855_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_856_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_857_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_858_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_859_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_860_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_861_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_862_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_863_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_864_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_865_one__add__one,axiom,
    aa_real_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_866_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_867_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_868_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_869_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_870_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_871_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_872_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_873_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_874_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_875_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_876_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_877_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_878_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_879_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_880_pred__member,axiom,
    ! [Ta: vEBT_VEBT,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),X3),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3)
        & ! [Z3: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y) ) ) ) ).

% pred_member
tff(fact_881_case4_I7_J,axiom,
    ! [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))),m))
     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList2,I2)),X_12)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary2),I2) ) ) ).

% case4(7)
tff(fact_882_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_883_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_884_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_885_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_886_lt__ex,axiom,
    ! [X3: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X3) ).

% lt_ex
tff(fact_887_lt__ex,axiom,
    ! [X3: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X3) ).

% lt_ex
tff(fact_888_lt__ex,axiom,
    ! [X3: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),X3) ).

% lt_ex
tff(fact_889_gt__ex,axiom,
    ! [X3: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_13) ).

% gt_ex
tff(fact_890_gt__ex,axiom,
    ! [X3: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_13) ).

% gt_ex
tff(fact_891_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),X_13) ).

% gt_ex
tff(fact_892_gt__ex,axiom,
    ! [X3: int] :
    ? [X_13: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X3),X_13) ).

% gt_ex
tff(fact_893_dense,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Y) ) ) ).

% dense
tff(fact_894_dense,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Y) ) ) ).

% dense
tff(fact_895_less__imp__neq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_896_less__imp__neq,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_897_less__imp__neq,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_898_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_899_less__imp__neq,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_900_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_901_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_902_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_903_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_904_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_905_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_906_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_907_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_908_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_909_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_910_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_911_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_912_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_913_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_914_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_915_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y5),X5)
             => aa_nat_o(P,Y5) )
         => aa_nat_o(P,X5) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_916_antisym__conv3,axiom,
    ! [Y: real,X3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_917_antisym__conv3,axiom,
    ! [Y: rat,X3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_918_antisym__conv3,axiom,
    ! [Y: num,X3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_919_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_920_antisym__conv3,axiom,
    ! [Y: int,X3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_921_linorder__cases,axiom,
    ! [X3: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( ( X3 != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ) ).

% linorder_cases
tff(fact_922_linorder__cases,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( ( X3 != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ) ).

% linorder_cases
tff(fact_923_linorder__cases,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( ( X3 != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ) ).

% linorder_cases
tff(fact_924_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( ( X3 != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ) ).

% linorder_cases
tff(fact_925_linorder__cases,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( ( X3 != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ) ).

% linorder_cases
tff(fact_926_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_927_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_928_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_929_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_930_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_931_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_932_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_933_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_934_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_935_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_936_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N3: nat] :
          ( aa_nat_o(P,N3)
          & ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N3)
             => ~ aa_nat_o(P,M4) ) ) ) ).

% exists_least_iff
tff(fact_937_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B3)
         => aa_real_o(aa_real_fun_real_o(P,A3),B3) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P,A3),A3)
       => ( ! [A3: real,B3: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B3),A3)
             => aa_real_o(aa_real_fun_real_o(P,A3),B3) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_938_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A3),A3)
       => ( ! [A3: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_939_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P,A3),A3)
       => ( ! [A3: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B3),A3)
             => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
         => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_940_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_941_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
       => ( ! [A3: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B3),A3)
             => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_942_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_943_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_944_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_945_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_946_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_947_not__less__iff__gr__or__eq,axiom,
    ! [X3: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3)
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_948_not__less__iff__gr__or__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3)
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_949_not__less__iff__gr__or__eq,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3)
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_950_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3)
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_951_not__less__iff__gr__or__eq,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3)
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_952_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_953_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_954_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_955_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_956_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_957_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_958_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_959_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_960_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_961_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_962_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_963_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_964_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_965_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_966_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_967_linorder__neqE,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ) ).

% linorder_neqE
tff(fact_968_linorder__neqE,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ) ).

% linorder_neqE
tff(fact_969_linorder__neqE,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ) ).

% linorder_neqE
tff(fact_970_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ) ).

% linorder_neqE
tff(fact_971_linorder__neqE,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ) ).

% linorder_neqE
tff(fact_972_order__less__asym,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ).

% order_less_asym
tff(fact_973_order__less__asym,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ).

% order_less_asym
tff(fact_974_order__less__asym,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ).

% order_less_asym
tff(fact_975_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ).

% order_less_asym
tff(fact_976_order__less__asym,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ).

% order_less_asym
tff(fact_977_linorder__neq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ) ).

% linorder_neq_iff
tff(fact_978_linorder__neq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ) ).

% linorder_neq_iff
tff(fact_979_linorder__neq__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ) ).

% linorder_neq_iff
tff(fact_980_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ) ).

% linorder_neq_iff
tff(fact_981_linorder__neq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ) ).

% linorder_neq_iff
tff(fact_982_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_983_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_984_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_985_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_986_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_987_order__less__trans,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z) ) ) ).

% order_less_trans
tff(fact_988_order__less__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z) ) ) ).

% order_less_trans
tff(fact_989_order__less__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z) ) ) ).

% order_less_trans
tff(fact_990_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z) ) ) ).

% order_less_trans
tff(fact_991_order__less__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z) ) ) ).

% order_less_trans
tff(fact_992_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_993_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_994_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X5)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_995_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_996_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X5)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_997_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_998_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_999_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1000_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1001_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1002_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1003_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1004_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X5)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1005_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1006_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X5)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1007_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1008_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1009_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1010_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1011_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1012_order__less__irrefl,axiom,
    ! [X3: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X3) ).

% order_less_irrefl
tff(fact_1013_order__less__irrefl,axiom,
    ! [X3: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X3) ).

% order_less_irrefl
tff(fact_1014_order__less__irrefl,axiom,
    ! [X3: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),X3) ).

% order_less_irrefl
tff(fact_1015_order__less__irrefl,axiom,
    ! [X3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),X3) ).

% order_less_irrefl
tff(fact_1016_order__less__irrefl,axiom,
    ! [X3: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),X3) ).

% order_less_irrefl
tff(fact_1017_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1018_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1019_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X5)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1020_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1021_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1022_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1023_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1024_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1025_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1026_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1027_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1028_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1029_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X5)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1030_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1031_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X5)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1032_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1033_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1034_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1035_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1036_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1037_order__less__not__sym,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ).

% order_less_not_sym
tff(fact_1038_order__less__not__sym,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ).

% order_less_not_sym
tff(fact_1039_order__less__not__sym,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ).

% order_less_not_sym
tff(fact_1040_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ).

% order_less_not_sym
tff(fact_1041_order__less__not__sym,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ).

% order_less_not_sym
tff(fact_1042_order__less__imp__triv,axiom,
    ! [X3: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1043_order__less__imp__triv,axiom,
    ! [X3: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1044_order__less__imp__triv,axiom,
    ! [X3: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1045_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1046_order__less__imp__triv,axiom,
    ! [X3: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1047_linorder__less__linear,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
      | ( X3 = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ).

% linorder_less_linear
tff(fact_1048_linorder__less__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
      | ( X3 = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ).

% linorder_less_linear
tff(fact_1049_linorder__less__linear,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
      | ( X3 = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ).

% linorder_less_linear
tff(fact_1050_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
      | ( X3 = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ).

% linorder_less_linear
tff(fact_1051_linorder__less__linear,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
      | ( X3 = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ).

% linorder_less_linear
tff(fact_1052_order__less__imp__not__eq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1053_order__less__imp__not__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1054_order__less__imp__not__eq,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1055_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1056_order__less__imp__not__eq,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1057_order__less__imp__not__eq2,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1058_order__less__imp__not__eq2,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1059_order__less__imp__not__eq2,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1060_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1061_order__less__imp__not__eq2,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1062_order__less__imp__not__less,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ).

% order_less_imp_not_less
tff(fact_1063_order__less__imp__not__less,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ).

% order_less_imp_not_less
tff(fact_1064_order__less__imp__not__less,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ).

% order_less_imp_not_less
tff(fact_1065_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ).

% order_less_imp_not_less
tff(fact_1066_order__less__imp__not__less,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ).

% order_less_imp_not_less
tff(fact_1067_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_1068_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_1069_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_1070_less__not__refl3,axiom,
    ! [Sb: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Sb),Ta)
     => ( Sb != Ta ) ) ).

% less_not_refl3
tff(fact_1071_one__reorient,axiom,
    ! [X3: complex] :
      ( ( one_one_complex = X3 )
    <=> ( X3 = one_one_complex ) ) ).

% one_reorient
tff(fact_1072_one__reorient,axiom,
    ! [X3: real] :
      ( ( one_one_real = X3 )
    <=> ( X3 = one_one_real ) ) ).

% one_reorient
tff(fact_1073_one__reorient,axiom,
    ! [X3: rat] :
      ( ( one_one_rat = X3 )
    <=> ( X3 = one_one_rat ) ) ).

% one_reorient
tff(fact_1074_one__reorient,axiom,
    ! [X3: nat] :
      ( ( one_one_nat = X3 )
    <=> ( X3 = one_one_nat ) ) ).

% one_reorient
tff(fact_1075_one__reorient,axiom,
    ! [X3: int] :
      ( ( one_one_int = X3 )
    <=> ( X3 = one_one_int ) ) ).

% one_reorient
tff(fact_1076_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_1077_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_1078_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_1079_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ) ).

% linorder_neqE_nat
tff(fact_1080_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_1081_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_1082_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_1083_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_1084_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_1085_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1086_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1087_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1088_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1089_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1090_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_1091_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_1092_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_1093_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_1094_order__le__imp__less__or__eq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1095_order__le__imp__less__or__eq,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_set_nat(X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1096_order__le__imp__less__or__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1097_order__le__imp__less__or__eq,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1098_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1099_order__le__imp__less__or__eq,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1100_linorder__le__less__linear,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ).

% linorder_le_less_linear
tff(fact_1101_linorder__le__less__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ).

% linorder_le_less_linear
tff(fact_1102_linorder__le__less__linear,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ).

% linorder_le_less_linear
tff(fact_1103_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ).

% linorder_le_less_linear
tff(fact_1104_linorder__le__less__linear,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ).

% linorder_le_less_linear
tff(fact_1105_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1106_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1107_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X5)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1108_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1109_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1110_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1111_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1112_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1113_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1114_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1115_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1116_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1117_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1118_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1119_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1120_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X5)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1121_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1122_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1123_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1124_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1125_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1126_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1127_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1128_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1129_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1130_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X5)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1131_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1132_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1133_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1134_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1135_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1136_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1137_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X5)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1138_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1139_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1140_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X5: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1141_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X5: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1142_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X5: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1143_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1144_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X5: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1145_order__less__le__trans,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z) ) ) ).

% order_less_le_trans
tff(fact_1146_order__less__le__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_set_nat(X3),Z) ) ) ).

% order_less_le_trans
tff(fact_1147_order__less__le__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z) ) ) ).

% order_less_le_trans
tff(fact_1148_order__less__le__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z) ) ) ).

% order_less_le_trans
tff(fact_1149_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z) ) ) ).

% order_less_le_trans
tff(fact_1150_order__less__le__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z) ) ) ).

% order_less_le_trans
tff(fact_1151_order__le__less__trans,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z) ) ) ).

% order_le_less_trans
tff(fact_1152_order__le__less__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_set_nat(X3),Z) ) ) ).

% order_le_less_trans
tff(fact_1153_order__le__less__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z) ) ) ).

% order_le_less_trans
tff(fact_1154_order__le__less__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z) ) ) ).

% order_le_less_trans
tff(fact_1155_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z) ) ) ).

% order_le_less_trans
tff(fact_1156_order__le__less__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z) ) ) ).

% order_le_less_trans
tff(fact_1157_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1158_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_neq_le_trans
tff(fact_1159_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1160_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1161_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1162_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1163_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1164_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( A != B )
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_le_neq_trans
tff(fact_1165_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1166_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1167_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1168_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1169_order__less__imp__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ).

% order_less_imp_le
tff(fact_1170_order__less__imp__le,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(X3),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(X3),Y) ) ).

% order_less_imp_le
tff(fact_1171_order__less__imp__le,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ).

% order_less_imp_le
tff(fact_1172_order__less__imp__le,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y) ) ).

% order_less_imp_le
tff(fact_1173_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ).

% order_less_imp_le
tff(fact_1174_order__less__imp__le,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ).

% order_less_imp_le
tff(fact_1175_linorder__not__less,axiom,
    ! [X3: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3) ) ).

% linorder_not_less
tff(fact_1176_linorder__not__less,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) ) ).

% linorder_not_less
tff(fact_1177_linorder__not__less,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) ) ).

% linorder_not_less
tff(fact_1178_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ).

% linorder_not_less
tff(fact_1179_linorder__not__less,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) ) ).

% linorder_not_less
tff(fact_1180_linorder__not__le,axiom,
    ! [X3: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3) ) ).

% linorder_not_le
tff(fact_1181_linorder__not__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X3) ) ).

% linorder_not_le
tff(fact_1182_linorder__not__le,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X3) ) ).

% linorder_not_le
tff(fact_1183_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ).

% linorder_not_le
tff(fact_1184_linorder__not__le,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ).

% linorder_not_le
tff(fact_1185_order__less__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1186_order__less__le,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(X3),Y)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1187_order__less__le,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1188_order__less__le,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1189_order__less__le,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1190_order__less__le,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1191_order__le__less,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1192_order__le__less,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
    <=> ( aa_set_nat_o(ord_less_set_nat(X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1193_order__le__less,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1194_order__le__less,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1195_order__le__less,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1196_order__le__less,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1197_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1198_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1199_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1200_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1201_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1202_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1203_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_1204_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% order.strict_implies_order
tff(fact_1205_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_1206_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_1207_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_1208_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_1209_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1210_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1211_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1212_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1213_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1214_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1215_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1216_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1217_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1218_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1219_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1220_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1221_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1222_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1223_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1224_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1225_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1226_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1227_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1228_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1229_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1230_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1231_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1232_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1233_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1234_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_set_nat(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1235_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1236_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1237_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1238_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1239_dense__le__bounded,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1240_dense__le__bounded,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1241_dense__ge__bounded,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X3)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),X3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1242_dense__ge__bounded,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),X3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1243_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1244_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% order.strict_iff_not
tff(fact_1245_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1246_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1247_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1248_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1249_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_1250_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans2
tff(fact_1251_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_1252_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_1253_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_1254_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_1255_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_1256_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans1
tff(fact_1257_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_1258_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_1259_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_1260_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_1261_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1262_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1263_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1264_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1265_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1266_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1267_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1268_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_set_nat(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1269_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1270_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1271_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1272_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1273_not__le__imp__less,axiom,
    ! [Y: real,X3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ).

% not_le_imp_less
tff(fact_1274_not__le__imp__less,axiom,
    ! [Y: rat,X3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y) ) ).

% not_le_imp_less
tff(fact_1275_not__le__imp__less,axiom,
    ! [Y: num,X3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y) ) ).

% not_le_imp_less
tff(fact_1276_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ).

% not_le_imp_less
tff(fact_1277_not__le__imp__less,axiom,
    ! [Y: int,X3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y) ) ).

% not_le_imp_less
tff(fact_1278_less__le__not__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3) ) ) ).

% less_le_not_le
tff(fact_1279_less__le__not__le,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(X3),Y)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(Y),X3) ) ) ).

% less_le_not_le
tff(fact_1280_less__le__not__le,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) ) ) ).

% less_le_not_le
tff(fact_1281_less__le__not__le,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) ) ) ).

% less_le_not_le
tff(fact_1282_less__le__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ) ).

% less_le_not_le
tff(fact_1283_less__le__not__le,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) ) ) ).

% less_le_not_le
tff(fact_1284_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_1285_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_1286_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X5) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_1287_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X5) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_1288_antisym__conv2,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1289_antisym__conv2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( ~ aa_set_nat_o(ord_less_set_nat(X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1290_antisym__conv2,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1291_antisym__conv2,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1292_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1293_antisym__conv2,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1294_antisym__conv1,axiom,
    ! [X3: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1295_antisym__conv1,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1296_antisym__conv1,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1297_antisym__conv1,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1298_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1299_antisym__conv1,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1300_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1301_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1302_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1303_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1304_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1305_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1306_leI,axiom,
    ! [X3: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3) ) ).

% leI
tff(fact_1307_leI,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) ) ).

% leI
tff(fact_1308_leI,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3) ) ).

% leI
tff(fact_1309_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ).

% leI
tff(fact_1310_leI,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3) ) ).

% leI
tff(fact_1311_leD,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ).

% leD
tff(fact_1312_leD,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
     => ~ aa_set_nat_o(ord_less_set_nat(X3),Y) ) ).

% leD
tff(fact_1313_leD,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y) ) ).

% leD
tff(fact_1314_leD,axiom,
    ! [Y: num,X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y) ) ).

% leD
tff(fact_1315_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ).

% leD
tff(fact_1316_leD,axiom,
    ! [Y: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y) ) ).

% leD
tff(fact_1317_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_1318_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1319_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1320_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_1321_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_1322_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1323_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1324_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_1325_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_strict_right_mono
tff(fact_1326_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_1327_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_1328_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_1329_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_strict_left_mono
tff(fact_1330_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_1331_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_1332_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_1333_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_strict_mono
tff(fact_1334_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_strict_mono
tff(fact_1335_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_1336_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_1337_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1338_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1339_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1340_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1341_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1342_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1343_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1344_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1345_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1346_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1347_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1348_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K2: 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,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1349_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,bot_bo2099793752762293965at_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1350_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_1351_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1352_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_1353_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_1354_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),bot_bo2099793752762293965at_nat) ).

% bot.extremum_strict
tff(fact_1355_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_1356_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1357_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_1358_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_1359_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1360_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(P,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I3))
               => aa_nat_o(P,I3) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_1361_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J2: nat,K: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_1362_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K2) ) ) ).

% less_trans_Suc
tff(fact_1363_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_1364_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_1365_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M5: nat] :
          ( ( Mb = aa_nat_nat(suc,M5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M5) ) ) ).

% Suc_less_eq2
tff(fact_1366_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
           => aa_nat_o(P,I4) ) ) ) ).

% All_less_Suc
tff(fact_1367_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_1368_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_1369_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
            & aa_nat_o(P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_1370_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_1371_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_1372_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_1373_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K2)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K2 != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1374_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_1375_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
     => ( ( K2 != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K2 != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1376_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),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_1377_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_1378_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_1379_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_1380_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_1381_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_1382_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_1383_trans__less__add2,axiom,
    ! [I: nat,J: nat,Mb: 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,Mb),J)) ) ).

% trans_less_add2
tff(fact_1384_trans__less__add1,axiom,
    ! [I: nat,J: nat,Mb: 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),Mb)) ) ).

% trans_less_add1
tff(fact_1385_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_less_mono1
tff(fact_1386_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_1387_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_1388_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_1389_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: 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)),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2) ) ).

% add_lessD1
tff(fact_1390_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_1391_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_1392_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_1393_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_1394_one__plus__numeral__commute,axiom,
    ! [X3: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X3)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X3)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_1395_one__plus__numeral__commute,axiom,
    ! [X3: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,X3)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,X3)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1396_one__plus__numeral__commute,axiom,
    ! [X3: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,X3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X3)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1397_one__plus__numeral__commute,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X3)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_1398_one__plus__numeral__commute,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X3)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_1399_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_1400_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_1401_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_1402_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_1403_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_1404_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_1405_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_less_le_mono
tff(fact_1406_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1407_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1408_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_1409_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_le_less_mono
tff(fact_1410_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1411_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1412_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_1413_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1414_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1415_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1416_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K2: 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,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1417_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1418_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1419_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1420_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K2: 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,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1421_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_1422_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_1423_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_1424_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_1425_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_1426_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_1427_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_1428_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_1429_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_1430_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)) ) ).

% less_imp_Suc_add
tff(fact_1431_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_1432_less__add__Suc2,axiom,
    ! [I: nat,Mb: 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,Mb),I))) ).

% less_add_Suc2
tff(fact_1433_less__add__Suc1,axiom,
    ! [I: nat,Mb: 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),Mb))) ).

% less_add_Suc1
tff(fact_1434_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q2: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ) ).

% less_natE
tff(fact_1435_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K2: nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K2)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2))) ) ).

% mono_nat_linear_lb
tff(fact_1436_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_1437_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1438_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1439_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_1440_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% is_num_normalize(1)
tff(fact_1441_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% is_num_normalize(1)
tff(fact_1442_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_1443_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).

% add_One_commute
tff(fact_1444_le__num__One__iff,axiom,
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),one)
    <=> ( X3 = one ) ) ).

% le_num_One_iff
tff(fact_1445_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_1446_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_1447_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_1448_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_1449_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_1450_Suc__nat__number__of__add,axiom,
    ! [V2: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V2)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_1451_pred__max,axiom,
    ! [Dega: nat,Maa: nat,X3: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),X3)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = some_nat(Maa) ) ) ) ).

% pred_max
tff(fact_1452_power__increasing__iff,axiom,
    ! [B: real,X3: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),X3)),aa_nat_real(power_power_real(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ).

% power_increasing_iff
tff(fact_1453_power__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X3)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ).

% power_increasing_iff
tff(fact_1454_power__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ).

% power_increasing_iff
tff(fact_1455_power__increasing__iff,axiom,
    ! [B: int,X3: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),X3)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ).

% power_increasing_iff
tff(fact_1456_greater__shift,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(X3)),some_nat(Y)) ) ).

% greater_shift
tff(fact_1457_less__shift,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(X3)),some_nat(Y)) ) ).

% less_shift
tff(fact_1458_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,X3) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpypredd
tff(fact_1459_power__strict__increasing__iff,axiom,
    ! [B: real,X3: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),X3)),aa_nat_real(power_power_real(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1460_power__strict__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),X3)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1461_power__strict__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1462_power__strict__increasing__iff,axiom,
    ! [B: int,X3: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),X3)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1463_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      vEBT_VEBT_insert(vEBT_Node(Infoa,Dega,TreeLista,Summarya),X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)),X3),vEBT_Node(Infoa,Dega,TreeLista,Summarya),vEBT_vebt_insert(vEBT_Node(Infoa,Dega,TreeLista,Summarya),X3)) ).

% VEBT_internal.insert'.simps(2)
tff(fact_1464_sumtreelistcong,axiom,
    ! [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))),m))
     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I2)),X_12)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I2) ) ) ).

% sumtreelistcong
tff(fact_1465_setcongy,axiom,
    ! [I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList2,I)) = vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,I)) ) ) ).

% setcongy
tff(fact_1466_membercongy,axiom,
    ! [I: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList2,I)),X3)
      <=> aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,I)),X3) ) ) ).

% membercongy
tff(fact_1467_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_1468_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),set_complex2(Xs))
         => aa_complex_o(P,X5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_1469_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(Xs))
         => aa_real_o(P,X5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_1470_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_1471_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),set_o2(Xs))
         => aa_o_o(P,(X5)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_1472_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
         => aa_nat_o(P,X5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_1473_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(Xs))
         => aa_int_o(P,X5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_1474_psubsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( ( A4 != B4 )
       => aa_set_nat_o(ord_less_set_nat(A4),B4) ) ) ).

% psubsetI
tff(fact_1475_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Px: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,X3) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),X3),Px) ) ) ).

% pred_corr
tff(fact_1476_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,X3) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Ta)),X3),Sx) ) ) ).

% pred_correct
tff(fact_1477_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_1478_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_1479_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(power_power_real(one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_1480_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_1481_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_1482_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_1483_power__one__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),one_one_nat) = A ).

% power_one_right
tff(fact_1484_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_1485_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_1486_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(power_power_real(A),Mb) = aa_nat_real(power_power_real(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_1487_power__inject__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_1488_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_1489_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_1490_treecongy,axiom,
    ! [I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( nth_VEBT_VEBT(treeList2,I) = nth_VEBT_VEBT(treeList,I) ) ) ).

% treecongy
tff(fact_1491_not__psubset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),bot_bo2099793752762293965at_nat) ).

% not_psubset_empty
tff(fact_1492_not__psubset__empty,axiom,
    ! [A4: set_o] : ~ aa_set_o_o(ord_less_set_o(A4),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1493_not__psubset__empty,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A4),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1494_not__psubset__empty,axiom,
    ! [A4: set_int] : ~ aa_set_int_o(ord_less_set_int(A4),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1495_psubsetE,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
         => aa_set_nat_o(ord_less_eq_set_nat(B4),A4) ) ) ).

% psubsetE
tff(fact_1496_psubset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
        & ( A4 != B4 ) ) ) ).

% psubset_eq
tff(fact_1497_psubset__imp__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ).

% psubset_imp_subset
tff(fact_1498_psubset__subset__trans,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C2)
       => aa_set_nat_o(ord_less_set_nat(A4),C2) ) ) ).

% psubset_subset_trans
tff(fact_1499_subset__not__subset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B4),A4) ) ) ).

% subset_not_subset_eq
tff(fact_1500_subset__psubset__trans,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(ord_less_set_nat(B4),C2)
       => aa_set_nat_o(ord_less_set_nat(A4),C2) ) ) ).

% subset_psubset_trans
tff(fact_1501_subset__iff__psubset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
    <=> ( aa_set_nat_o(ord_less_set_nat(A4),B4)
        | ( A4 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1502_one__le__power,axiom,
    ! [A: real,Nb: 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(power_power_real(A),Nb)) ) ).

% one_le_power
tff(fact_1503_one__le__power,axiom,
    ! [A: rat,Nb: 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),Nb)) ) ).

% one_le_power
tff(fact_1504_one__le__power,axiom,
    ! [A: nat,Nb: 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),Nb)) ) ).

% one_le_power
tff(fact_1505_one__le__power,axiom,
    ! [A: int,Nb: 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),Nb)) ) ).

% one_le_power
tff(fact_1506_power__gt1,axiom,
    ! [A: real,Nb: 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(power_power_real(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1507_power__gt1,axiom,
    ! [A: rat,Nb: 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,Nb))) ) ).

% power_gt1
tff(fact_1508_power__gt1,axiom,
    ! [A: nat,Nb: 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,Nb))) ) ).

% power_gt1
tff(fact_1509_power__gt1,axiom,
    ! [A: int,Nb: 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,Nb))) ) ).

% power_gt1
tff(fact_1510_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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(power_power_real(A),Nb)),aa_nat_real(power_power_real(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_1511_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_1512_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_1513_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_1514_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Nb: 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(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1515_power__less__imp__less__exp,axiom,
    ! [A: rat,Mb: nat,Nb: 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),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1516_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1517_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Nb: 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),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1518_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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(power_power_real(A),Nb)),aa_nat_real(power_power_real(A),N4)) ) ) ).

% power_increasing
tff(fact_1519_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_increasing
tff(fact_1520_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_1521_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_increasing
tff(fact_1522_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: 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(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1523_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Nb: 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),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1524_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1525_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: 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),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1526_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_1527_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_1528_one__power2,axiom,
    aa_nat_real(power_power_real(one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_1529_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_1530_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_1531_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% less_exp
tff(fact_1532_self__le__ge2__pow,axiom,
    ! [K2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)) ) ).

% self_le_ge2_pow
tff(fact_1533_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_1534_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_1535_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_1536_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_1537_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_1538_insert__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(Ta)),insert_nat(X3,bot_bot_set_nat)) = vEBT_set_vebt(vEBT_vebt_insert(Ta,X3)) ) ) ) ).

% insert_correct
tff(fact_1539_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_1540_insert__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_VEBT_set_vebt(Ta)),insert_nat(X3,bot_bot_set_nat)) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(Ta,X3)) ) ) ) ).

% insert_corr
tff(fact_1541_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_1542_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).

% semiring_norm(75)
tff(fact_1543_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_1544_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_1545_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_1546_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_1547_UnCI,axiom,
    ! [C: complex,B4: set_complex,A4: set_complex] :
      ( ( ~ aa_set_complex_o(member_complex(C),B4)
       => aa_set_complex_o(member_complex(C),A4) )
     => aa_set_complex_o(member_complex(C),sup_sup_set_complex(A4,B4)) ) ).

% UnCI
tff(fact_1548_UnCI,axiom,
    ! [C: real,B4: set_real,A4: set_real] :
      ( ( ~ aa_set_real_o(member_real(C),B4)
       => aa_set_real_o(member_real(C),A4) )
     => aa_set_real_o(member_real(C),sup_sup_set_real(A4,B4)) ) ).

% UnCI
tff(fact_1549_UnCI,axiom,
    ! [C: $o,B4: set_o,A4: set_o] :
      ( ( ~ aa_set_o_o(member_o((C)),B4)
       => aa_set_o_o(member_o((C)),A4) )
     => aa_set_o_o(member_o((C)),sup_sup_set_o(A4,B4)) ) ).

% UnCI
tff(fact_1550_UnCI,axiom,
    ! [C: int,B4: set_int,A4: set_int] :
      ( ( ~ aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),A4) )
     => aa_set_int_o(member_int(C),sup_sup_set_int(A4,B4)) ) ).

% UnCI
tff(fact_1551_UnCI,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),A4) )
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ) ).

% UnCI
tff(fact_1552_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B4)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A4) )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ) ).

% UnCI
tff(fact_1553_Un__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),sup_sup_set_complex(A4,B4))
    <=> ( aa_set_complex_o(member_complex(C),A4)
        | aa_set_complex_o(member_complex(C),B4) ) ) ).

% Un_iff
tff(fact_1554_Un__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),sup_sup_set_real(A4,B4))
    <=> ( aa_set_real_o(member_real(C),A4)
        | aa_set_real_o(member_real(C),B4) ) ) ).

% Un_iff
tff(fact_1555_Un__iff,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),sup_sup_set_o(A4,B4))
    <=> ( aa_set_o_o(member_o((C)),A4)
        | aa_set_o_o(member_o((C)),B4) ) ) ).

% Un_iff
tff(fact_1556_Un__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),sup_sup_set_int(A4,B4))
    <=> ( aa_set_int_o(member_int(C),A4)
        | aa_set_int_o(member_int(C),B4) ) ) ).

% Un_iff
tff(fact_1557_Un__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
    <=> ( aa_set_nat_o(member_nat(C),A4)
        | aa_set_nat_o(member_nat(C),B4) ) ) ).

% Un_iff
tff(fact_1558_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A4)
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B4) ) ) ).

% Un_iff
tff(fact_1559_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_1560_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa_num_num(bit0,Mb) != one ).

% semiring_norm(85)
tff(fact_1561_Un__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = bot_bo228742789529271731at_nat )
    <=> ( ( A4 = bot_bo228742789529271731at_nat )
        & ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
tff(fact_1562_Un__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> ( ( A4 = bot_bo2099793752762293965at_nat )
        & ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
tff(fact_1563_Un__empty,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( sup_sup_set_o(A4,B4) = bot_bot_set_o )
    <=> ( ( A4 = bot_bot_set_o )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_1564_Un__empty,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = bot_bot_set_nat )
    <=> ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_1565_Un__empty,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( sup_sup_set_int(A4,B4) = bot_bot_set_int )
    <=> ( ( A4 = bot_bot_set_int )
        & ( B4 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_1566_Un__subset__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C2)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),C2)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),C2) ) ) ).

% Un_subset_iff
tff(fact_1567_Un__subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A4),C2)
        & aa_set_nat_o(ord_less_eq_set_nat(B4),C2) ) ) ).

% Un_subset_iff
tff(fact_1568_Un__insert__right,axiom,
    ! [A4: set_int,A: int,B4: set_int] : sup_sup_set_int(A4,insert_int(A,B4)) = insert_int(A,sup_sup_set_int(A4,B4)) ).

% Un_insert_right
tff(fact_1569_Un__insert__right,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : sup_sup_set_o(A4,insert_o((A),B4)) = insert_o((A),sup_sup_set_o(A4,B4)) ).

% Un_insert_right
tff(fact_1570_Un__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),insert8211810215607154385at_nat(A,B4)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) ).

% Un_insert_right
tff(fact_1571_Un__insert__right,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),insert_nat(A,B4)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ).

% Un_insert_right
tff(fact_1572_Un__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),insert9069300056098147895at_nat(A,B4)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ).

% Un_insert_right
tff(fact_1573_Un__insert__left,axiom,
    ! [A: int,B4: set_int,C2: set_int] : sup_sup_set_int(insert_int(A,B4),C2) = insert_int(A,sup_sup_set_int(B4,C2)) ).

% Un_insert_left
tff(fact_1574_Un__insert__left,axiom,
    ! [A: $o,B4: set_o,C2: set_o] : sup_sup_set_o(insert_o((A),B4),C2) = insert_o((A),sup_sup_set_o(B4,C2)) ).

% Un_insert_left
tff(fact_1575_Un__insert__left,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A,B4)),C2) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C2)) ).

% Un_insert_left
tff(fact_1576_Un__insert__left,axiom,
    ! [A: nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,B4)),C2) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) ).

% Un_insert_left
tff(fact_1577_Un__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A,B4)),C2) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) ).

% Un_insert_left
tff(fact_1578_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(6)
tff(fact_1579_psubsetD,axiom,
    ! [A4: set_complex,B4: set_complex,C: complex] :
      ( aa_set_complex_o(ord_less_set_complex(A4),B4)
     => ( aa_set_complex_o(member_complex(C),A4)
       => aa_set_complex_o(member_complex(C),B4) ) ) ).

% psubsetD
tff(fact_1580_psubsetD,axiom,
    ! [A4: set_real,B4: set_real,C: real] :
      ( aa_set_real_o(ord_less_set_real(A4),B4)
     => ( aa_set_real_o(member_real(C),A4)
       => aa_set_real_o(member_real(C),B4) ) ) ).

% psubsetD
tff(fact_1581_psubsetD,axiom,
    ! [A4: set_o,B4: set_o,C: $o] :
      ( aa_set_o_o(ord_less_set_o(A4),B4)
     => ( aa_set_o_o(member_o((C)),A4)
       => aa_set_o_o(member_o((C)),B4) ) ) ).

% psubsetD
tff(fact_1582_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
     => ( aa_set_nat_o(member_nat(C),A4)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% psubsetD
tff(fact_1583_psubsetD,axiom,
    ! [A4: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ( aa_set_int_o(member_int(C),A4)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% psubsetD
tff(fact_1584_UnE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),sup_sup_set_complex(A4,B4))
     => ( ~ aa_set_complex_o(member_complex(C),A4)
       => aa_set_complex_o(member_complex(C),B4) ) ) ).

% UnE
tff(fact_1585_UnE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),sup_sup_set_real(A4,B4))
     => ( ~ aa_set_real_o(member_real(C),A4)
       => aa_set_real_o(member_real(C),B4) ) ) ).

% UnE
tff(fact_1586_UnE,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),sup_sup_set_o(A4,B4))
     => ( ~ aa_set_o_o(member_o((C)),A4)
       => aa_set_o_o(member_o((C)),B4) ) ) ).

% UnE
tff(fact_1587_UnE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),sup_sup_set_int(A4,B4))
     => ( ~ aa_set_int_o(member_int(C),A4)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% UnE
tff(fact_1588_UnE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( ~ aa_set_nat_o(member_nat(C),A4)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% UnE
tff(fact_1589_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A4)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B4) ) ) ).

% UnE
tff(fact_1590_UnI1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),A4)
     => aa_set_complex_o(member_complex(C),sup_sup_set_complex(A4,B4)) ) ).

% UnI1
tff(fact_1591_UnI1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),A4)
     => aa_set_real_o(member_real(C),sup_sup_set_real(A4,B4)) ) ).

% UnI1
tff(fact_1592_UnI1,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),A4)
     => aa_set_o_o(member_o((C)),sup_sup_set_o(A4,B4)) ) ).

% UnI1
tff(fact_1593_UnI1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A4)
     => aa_set_int_o(member_int(C),sup_sup_set_int(A4,B4)) ) ).

% UnI1
tff(fact_1594_UnI1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A4)
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ) ).

% UnI1
tff(fact_1595_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),A4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ) ).

% UnI1
tff(fact_1596_UnI2,axiom,
    ! [C: complex,B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(C),B4)
     => aa_set_complex_o(member_complex(C),sup_sup_set_complex(A4,B4)) ) ).

% UnI2
tff(fact_1597_UnI2,axiom,
    ! [C: real,B4: set_real,A4: set_real] :
      ( aa_set_real_o(member_real(C),B4)
     => aa_set_real_o(member_real(C),sup_sup_set_real(A4,B4)) ) ).

% UnI2
tff(fact_1598_UnI2,axiom,
    ! [C: $o,B4: set_o,A4: set_o] :
      ( aa_set_o_o(member_o((C)),B4)
     => aa_set_o_o(member_o((C)),sup_sup_set_o(A4,B4)) ) ).

% UnI2
tff(fact_1599_UnI2,axiom,
    ! [C: int,B4: set_int,A4: set_int] :
      ( aa_set_int_o(member_int(C),B4)
     => aa_set_int_o(member_int(C),sup_sup_set_int(A4,B4)) ) ).

% UnI2
tff(fact_1600_UnI2,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C),B4)
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ) ).

% UnI2
tff(fact_1601_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),B4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ) ).

% UnI2
tff(fact_1602_bex__Un,axiom,
    ! [A4: set_nat,B4: set_nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
          & aa_nat_o(P,X4) )
    <=> ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_nat_o(P,X4) )
        | ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
            & aa_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_1603_bex__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,P: fun_Pr3059359313727237556_nat_o] :
      ( ? [X4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
          & aa_Pro5947980084102686253_nat_o(P,X4) )
    <=> ( ? [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A4)
            & aa_Pro5947980084102686253_nat_o(P,X4) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B4)
            & aa_Pro5947980084102686253_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_1604_ball__Un,axiom,
    ! [A4: set_nat,B4: set_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
         => aa_nat_o(P,X4) )
    <=> ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_nat_o(P,X4) )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
           => aa_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_1605_ball__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,P: fun_Pr3059359313727237556_nat_o] :
      ( ! [X4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
         => aa_Pro5947980084102686253_nat_o(P,X4) )
    <=> ( ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A4)
           => aa_Pro5947980084102686253_nat_o(P,X4) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B4)
           => aa_Pro5947980084102686253_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_1606_Un__assoc,axiom,
    ! [A4: set_nat,B4: 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,A4),B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) ).

% Un_assoc
tff(fact_1607_Un__assoc,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) ).

% Un_assoc
tff(fact_1608_Un__absorb,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),A4) = A4 ).

% Un_absorb
tff(fact_1609_Un__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),A4) = A4 ).

% Un_absorb
tff(fact_1610_Un__commute,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A4) ).

% Un_commute
tff(fact_1611_Un__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),A4) ).

% Un_commute
tff(fact_1612_Un__left__absorb,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) ).

% Un_left_absorb
tff(fact_1613_Un__left__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) ).

% Un_left_absorb
tff(fact_1614_Un__left__commute,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),C2)) ).

% Un_left_commute
tff(fact_1615_Un__left__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),C2)) ).

% Un_left_commute
tff(fact_1616_Un__empty__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),bot_bo228742789529271731at_nat) = A4 ).

% Un_empty_right
tff(fact_1617_Un__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),bot_bo2099793752762293965at_nat) = A4 ).

% Un_empty_right
tff(fact_1618_Un__empty__right,axiom,
    ! [A4: set_o] : sup_sup_set_o(A4,bot_bot_set_o) = A4 ).

% Un_empty_right
tff(fact_1619_Un__empty__right,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),bot_bot_set_nat) = A4 ).

% Un_empty_right
tff(fact_1620_Un__empty__right,axiom,
    ! [A4: set_int] : sup_sup_set_int(A4,bot_bot_set_int) = A4 ).

% Un_empty_right
tff(fact_1621_Un__empty__left,axiom,
    ! [B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1622_Un__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1623_Un__empty__left,axiom,
    ! [B4: set_o] : sup_sup_set_o(bot_bot_set_o,B4) = B4 ).

% Un_empty_left
tff(fact_1624_Un__empty__left,axiom,
    ! [B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1625_Un__empty__left,axiom,
    ! [B4: set_int] : sup_sup_set_int(bot_bot_set_int,B4) = B4 ).

% Un_empty_left
tff(fact_1626_subset__Un__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_1627_subset__Un__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_1628_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),A4)
           => ! [B6: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B6),B4)
               => ( C2 != aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_1629_subset__UnE,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ~ ! [A5: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(A5),A4)
           => ! [B6: set_nat] :
                ( aa_set_nat_o(ord_less_eq_set_nat(B6),B4)
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_1630_Un__absorb2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = A4 ) ) ).

% Un_absorb2
tff(fact_1631_Un__absorb2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = A4 ) ) ).

% Un_absorb2
tff(fact_1632_Un__absorb1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_1633_Un__absorb1,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_1634_Un__upper2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ).

% Un_upper2
tff(fact_1635_Un__upper2,axiom,
    ! [B4: set_nat,A4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ).

% Un_upper2
tff(fact_1636_Un__upper1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ).

% Un_upper1
tff(fact_1637_Un__upper1,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ).

% Un_upper1
tff(fact_1638_Un__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),C2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),C2)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C2) ) ) ).

% Un_least
tff(fact_1639_Un__least,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C2) ) ) ).

% Un_least
tff(fact_1640_Un__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),C2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),D2)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D2)) ) ) ).

% Un_mono
tff(fact_1641_Un__mono,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat,D2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),D2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D2)) ) ) ).

% Un_mono
tff(fact_1642_singleton__Un__iff,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) )
        | ( ( A4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) )
          & ( B4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1643_singleton__Un__iff,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) )
        | ( ( A4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) )
          & ( B4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1644_singleton__Un__iff,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ( insert_o((X3),bot_bot_set_o) = sup_sup_set_o(A4,B4) )
    <=> ( ( ( A4 = bot_bot_set_o )
          & ( B4 = insert_o((X3),bot_bot_set_o) ) )
        | ( ( A4 = insert_o((X3),bot_bot_set_o) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A4 = insert_o((X3),bot_bot_set_o) )
          & ( B4 = insert_o((X3),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1645_singleton__Un__iff,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ( insert_nat(X3,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bot_set_nat )
          & ( B4 = insert_nat(X3,bot_bot_set_nat) ) )
        | ( ( A4 = insert_nat(X3,bot_bot_set_nat) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A4 = insert_nat(X3,bot_bot_set_nat) )
          & ( B4 = insert_nat(X3,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1646_singleton__Un__iff,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ( insert_int(X3,bot_bot_set_int) = sup_sup_set_int(A4,B4) )
    <=> ( ( ( A4 = bot_bot_set_int )
          & ( B4 = insert_int(X3,bot_bot_set_int) ) )
        | ( ( A4 = insert_int(X3,bot_bot_set_int) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A4 = insert_int(X3,bot_bot_set_int) )
          & ( B4 = insert_int(X3,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1647_Un__singleton__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) )
    <=> ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) )
        | ( ( A4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) )
          & ( B4 = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1648_Un__singleton__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) )
    <=> ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) )
        | ( ( A4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) )
          & ( B4 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1649_Un__singleton__iff,axiom,
    ! [A4: set_o,B4: set_o,X3: $o] :
      ( ( sup_sup_set_o(A4,B4) = insert_o((X3),bot_bot_set_o) )
    <=> ( ( ( A4 = bot_bot_set_o )
          & ( B4 = insert_o((X3),bot_bot_set_o) ) )
        | ( ( A4 = insert_o((X3),bot_bot_set_o) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A4 = insert_o((X3),bot_bot_set_o) )
          & ( B4 = insert_o((X3),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1650_Un__singleton__iff,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = insert_nat(X3,bot_bot_set_nat) )
    <=> ( ( ( A4 = bot_bot_set_nat )
          & ( B4 = insert_nat(X3,bot_bot_set_nat) ) )
        | ( ( A4 = insert_nat(X3,bot_bot_set_nat) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A4 = insert_nat(X3,bot_bot_set_nat) )
          & ( B4 = insert_nat(X3,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1651_Un__singleton__iff,axiom,
    ! [A4: set_int,B4: set_int,X3: int] :
      ( ( sup_sup_set_int(A4,B4) = insert_int(X3,bot_bot_set_int) )
    <=> ( ( ( A4 = bot_bot_set_int )
          & ( B4 = insert_int(X3,bot_bot_set_int) ) )
        | ( ( A4 = insert_int(X3,bot_bot_set_int) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A4 = insert_int(X3,bot_bot_set_int) )
          & ( B4 = insert_int(X3,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1652_insert__is__Un,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)),A4) ).

% insert_is_Un
tff(fact_1653_insert__is__Un,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),A4) ).

% insert_is_Un
tff(fact_1654_insert__is__Un,axiom,
    ! [A: $o,A4: set_o] : insert_o((A),A4) = sup_sup_set_o(insert_o((A),bot_bot_set_o),A4) ).

% insert_is_Un
tff(fact_1655_insert__is__Un,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,bot_bot_set_nat)),A4) ).

% insert_is_Un
tff(fact_1656_insert__is__Un,axiom,
    ! [A: int,A4: set_int] : insert_int(A,A4) = sup_sup_set_int(insert_int(A,bot_bot_set_int),A4) ).

% insert_is_Un
tff(fact_1657_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,X3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_set_vebt(Ta)),insert_nat(X3,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_1658_case4_I11_J,axiom,
    ( ( 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))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I2 )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [X: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),na) = I2 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),ma) ) ) ) ) ) ).

% case4(11)
tff(fact_1659_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_1660_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),bot_bo228742789529271731at_nat) = A ).

% sup_bot.right_neutral
tff(fact_1661_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),bot_bo2099793752762293965at_nat) = A ).

% sup_bot.right_neutral
tff(fact_1662_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] : sup_sup_set_o(A,bot_bot_set_o) = A ).

% sup_bot.right_neutral
tff(fact_1663_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_1664_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : sup_sup_set_int(A,bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_1665_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1666_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1667_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o = sup_sup_set_o(A,B) )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1668_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1669_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = sup_sup_set_int(A,B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1670_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_1671_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_1672_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] : sup_sup_set_o(bot_bot_set_o,A) = A ).

% sup_bot.left_neutral
tff(fact_1673_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_1674_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : sup_sup_set_int(bot_bot_set_int,A) = A ).

% sup_bot.left_neutral
tff(fact_1675_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1676_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1677_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( sup_sup_set_o(A,B) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1678_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1679_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( sup_sup_set_int(A,B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1680_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y) = bot_bo228742789529271731at_nat )
    <=> ( ( X3 = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1681_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Y) = bot_bo2099793752762293965at_nat )
    <=> ( ( X3 = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1682_sup__eq__bot__iff,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( sup_sup_set_o(X3,Y) = bot_bot_set_o )
    <=> ( ( X3 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_1683_sup__eq__bot__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = bot_bot_set_nat )
    <=> ( ( X3 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1684_sup__eq__bot__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( sup_sup_set_int(X3,Y) = bot_bot_set_int )
    <=> ( ( X3 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_1685_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y) )
    <=> ( ( X3 = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1686_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Y) )
    <=> ( ( X3 = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1687_bot__eq__sup__iff,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( bot_bot_set_o = sup_sup_set_o(X3,Y) )
    <=> ( ( X3 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_1688_bot__eq__sup__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) )
    <=> ( ( X3 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1689_bot__eq__sup__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( bot_bot_set_int = sup_sup_set_int(X3,Y) )
    <=> ( ( X3 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_1690_sup__bot__right,axiom,
    ! [X3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),bot_bo228742789529271731at_nat) = X3 ).

% sup_bot_right
tff(fact_1691_sup__bot__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),bot_bo2099793752762293965at_nat) = X3 ).

% sup_bot_right
tff(fact_1692_sup__bot__right,axiom,
    ! [X3: set_o] : sup_sup_set_o(X3,bot_bot_set_o) = X3 ).

% sup_bot_right
tff(fact_1693_sup__bot__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),bot_bot_set_nat) = X3 ).

% sup_bot_right
tff(fact_1694_sup__bot__right,axiom,
    ! [X3: set_int] : sup_sup_set_int(X3,bot_bot_set_int) = X3 ).

% sup_bot_right
tff(fact_1695_bit__split__inv,axiom,
    ! [X3: 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,X3),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),D)),D) = X3 ).

% bit_split_inv
tff(fact_1696_Diff__idemp,axiom,
    ! [A4: set_nat,B4: 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,A4),B4)),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) ).

% Diff_idemp
tff(fact_1697_Diff__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A4,B4))
    <=> ( aa_set_complex_o(member_complex(C),A4)
        & ~ aa_set_complex_o(member_complex(C),B4) ) ) ).

% Diff_iff
tff(fact_1698_Diff__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A4,B4))
    <=> ( aa_set_real_o(member_real(C),A4)
        & ~ aa_set_real_o(member_real(C),B4) ) ) ).

% Diff_iff
tff(fact_1699_Diff__iff,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A4,B4))
    <=> ( aa_set_o_o(member_o((C)),A4)
        & ~ aa_set_o_o(member_o((C)),B4) ) ) ).

% Diff_iff
tff(fact_1700_Diff__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A4,B4))
    <=> ( aa_set_int_o(member_int(C),A4)
        & ~ aa_set_int_o(member_int(C),B4) ) ) ).

% Diff_iff
tff(fact_1701_Diff__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
    <=> ( aa_set_nat_o(member_nat(C),A4)
        & ~ aa_set_nat_o(member_nat(C),B4) ) ) ).

% Diff_iff
tff(fact_1702_DiffI,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),A4)
     => ( ~ aa_set_complex_o(member_complex(C),B4)
       => aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A4,B4)) ) ) ).

% DiffI
tff(fact_1703_DiffI,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),A4)
     => ( ~ aa_set_real_o(member_real(C),B4)
       => aa_set_real_o(member_real(C),minus_minus_set_real(A4,B4)) ) ) ).

% DiffI
tff(fact_1704_DiffI,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),A4)
     => ( ~ aa_set_o_o(member_o((C)),B4)
       => aa_set_o_o(member_o((C)),minus_minus_set_o(A4,B4)) ) ) ).

% DiffI
tff(fact_1705_DiffI,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A4)
     => ( ~ aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),minus_minus_set_int(A4,B4)) ) ) ).

% DiffI
tff(fact_1706_DiffI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A4)
     => ( ~ aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) ) ) ).

% DiffI
tff(fact_1707_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_1708_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,X3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_VEBT_set_vebt(Ta)),insert_nat(X3,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_1709_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel
tff(fact_1710_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_1711_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_1712_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ).

% diff_add_cancel
tff(fact_1713_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ).

% diff_add_cancel
tff(fact_1714_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_1715_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_1716_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_left
tff(fact_1717_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_1718_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_1719_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1720_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1721_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1722_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1723_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_1724_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_right
tff(fact_1725_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_1726_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_1727_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1728_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1729_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1730_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1731_le__sup__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y)),Z)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),Z)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),Z) ) ) ).

% le_sup_iff
tff(fact_1732_le__sup__iff,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)),Z)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(X3),Z)
        & aa_set_nat_o(ord_less_eq_set_nat(Y),Z) ) ) ).

% le_sup_iff
tff(fact_1733_le__sup__iff,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1734_le__sup__iff,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1735_le__sup__iff,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1736_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A) ) ) ).

% sup.bounded_iff
tff(fact_1737_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% sup.bounded_iff
tff(fact_1738_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1739_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1740_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1741_sup__bot__left,axiom,
    ! [X3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),X3) = X3 ).

% sup_bot_left
tff(fact_1742_sup__bot__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),X3) = X3 ).

% sup_bot_left
tff(fact_1743_sup__bot__left,axiom,
    ! [X3: set_o] : sup_sup_set_o(bot_bot_set_o,X3) = X3 ).

% sup_bot_left
tff(fact_1744_sup__bot__left,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X3) = X3 ).

% sup_bot_left
tff(fact_1745_sup__bot__left,axiom,
    ! [X3: set_int] : sup_sup_set_int(bot_bot_set_int,X3) = X3 ).

% sup_bot_left
tff(fact_1746_Diff__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),bot_bo2099793752762293965at_nat) = A4 ).

% Diff_empty
tff(fact_1747_Diff__empty,axiom,
    ! [A4: set_o] : minus_minus_set_o(A4,bot_bot_set_o) = A4 ).

% Diff_empty
tff(fact_1748_Diff__empty,axiom,
    ! [A4: set_int] : minus_minus_set_int(A4,bot_bot_set_int) = A4 ).

% Diff_empty
tff(fact_1749_Diff__empty,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),bot_bot_set_nat) = A4 ).

% Diff_empty
tff(fact_1750_empty__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bot_bo2099793752762293965at_nat),A4) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_1751_empty__Diff,axiom,
    ! [A4: set_o] : minus_minus_set_o(bot_bot_set_o,A4) = bot_bot_set_o ).

% empty_Diff
tff(fact_1752_empty__Diff,axiom,
    ! [A4: set_int] : minus_minus_set_int(bot_bot_set_int,A4) = bot_bot_set_int ).

% empty_Diff
tff(fact_1753_empty__Diff,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A4) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1754_Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),A4) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_1755_Diff__cancel,axiom,
    ! [A4: set_o] : minus_minus_set_o(A4,A4) = bot_bot_set_o ).

% Diff_cancel
tff(fact_1756_Diff__cancel,axiom,
    ! [A4: set_int] : minus_minus_set_int(A4,A4) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1757_Diff__cancel,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),A4) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1758_Diff__insert0,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X3,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) ) ) ).

% Diff_insert0
tff(fact_1759_Diff__insert0,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X3,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4) ) ) ).

% Diff_insert0
tff(fact_1760_Diff__insert0,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(X3),A4)
     => ( minus_811609699411566653omplex(A4,insert_complex(X3,B4)) = minus_811609699411566653omplex(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1761_Diff__insert0,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(X3),A4)
     => ( minus_minus_set_real(A4,insert_real(X3,B4)) = minus_minus_set_real(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1762_Diff__insert0,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X3)),A4)
     => ( minus_minus_set_o(A4,insert_o((X3),B4)) = minus_minus_set_o(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1763_Diff__insert0,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X3),A4)
     => ( minus_minus_set_int(A4,insert_int(X3,B4)) = minus_minus_set_int(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1764_Diff__insert0,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X3),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) ) ) ).

% Diff_insert0
tff(fact_1765_insert__Diff1,axiom,
    ! [X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X3,A4)),B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) ) ) ).

% insert_Diff1
tff(fact_1766_insert__Diff1,axiom,
    ! [X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),B4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X3,A4)),B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4) ) ) ).

% insert_Diff1
tff(fact_1767_insert__Diff1,axiom,
    ! [X3: complex,B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(X3),B4)
     => ( minus_811609699411566653omplex(insert_complex(X3,A4),B4) = minus_811609699411566653omplex(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1768_insert__Diff1,axiom,
    ! [X3: real,B4: set_real,A4: set_real] :
      ( aa_set_real_o(member_real(X3),B4)
     => ( minus_minus_set_real(insert_real(X3,A4),B4) = minus_minus_set_real(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1769_insert__Diff1,axiom,
    ! [X3: $o,B4: set_o,A4: set_o] :
      ( aa_set_o_o(member_o((X3)),B4)
     => ( minus_minus_set_o(insert_o((X3),A4),B4) = minus_minus_set_o(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1770_insert__Diff1,axiom,
    ! [X3: int,B4: set_int,A4: set_int] :
      ( aa_set_int_o(member_int(X3),B4)
     => ( minus_minus_set_int(insert_int(X3,A4),B4) = minus_minus_set_int(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1771_insert__Diff1,axiom,
    ! [X3: nat,B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(X3),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X3,A4)),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) ) ) ).

% insert_Diff1
tff(fact_1772_Un__Diff__cancel2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),A4)),A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),A4) ).

% Un_Diff_cancel2
tff(fact_1773_Un__Diff__cancel2,axiom,
    ! [B4: set_nat,A4: 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,B4),A4)),A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A4) ).

% Un_Diff_cancel2
tff(fact_1774_Un__Diff__cancel,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),A4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) ).

% Un_Diff_cancel
tff(fact_1775_Un__Diff__cancel,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) ).

% Un_Diff_cancel
tff(fact_1776_Diff__eq__empty__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1777_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A4,B4) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1778_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A4,B4) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1779_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1780_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A,A4) ).

% insert_Diff_single
tff(fact_1781_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A4) ).

% insert_Diff_single
tff(fact_1782_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] : insert_o((A),minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = insert_o((A),A4) ).

% insert_Diff_single
tff(fact_1783_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] : insert_int(A,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = insert_int(A,A4) ).

% insert_Diff_single
tff(fact_1784_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A4) ).

% insert_Diff_single
tff(fact_1785_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_1786_acd,axiom,
    ( ( 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))),m))
       => ( ( ( 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)) )
          & ! [X: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),na) = I2 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),ma) ) ) ) ) ) ).

% acd
tff(fact_1787_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) ).

% diff_right_commute
tff(fact_1788_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) ).

% diff_right_commute
tff(fact_1789_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_1790_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_1791_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1792_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1793_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1794_DiffD2,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A4,B4))
     => ~ aa_set_complex_o(member_complex(C),B4) ) ).

% DiffD2
tff(fact_1795_DiffD2,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A4,B4))
     => ~ aa_set_real_o(member_real(C),B4) ) ).

% DiffD2
tff(fact_1796_DiffD2,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A4,B4))
     => ~ aa_set_o_o(member_o((C)),B4) ) ).

% DiffD2
tff(fact_1797_DiffD2,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A4,B4))
     => ~ aa_set_int_o(member_int(C),B4) ) ).

% DiffD2
tff(fact_1798_DiffD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
     => ~ aa_set_nat_o(member_nat(C),B4) ) ).

% DiffD2
tff(fact_1799_DiffD1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A4,B4))
     => aa_set_complex_o(member_complex(C),A4) ) ).

% DiffD1
tff(fact_1800_DiffD1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A4,B4))
     => aa_set_real_o(member_real(C),A4) ) ).

% DiffD1
tff(fact_1801_DiffD1,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A4,B4))
     => aa_set_o_o(member_o((C)),A4) ) ).

% DiffD1
tff(fact_1802_DiffD1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A4,B4))
     => aa_set_int_o(member_int(C),A4) ) ).

% DiffD1
tff(fact_1803_DiffD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
     => aa_set_nat_o(member_nat(C),A4) ) ).

% DiffD1
tff(fact_1804_DiffE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A4,B4))
     => ~ ( aa_set_complex_o(member_complex(C),A4)
         => aa_set_complex_o(member_complex(C),B4) ) ) ).

% DiffE
tff(fact_1805_DiffE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A4,B4))
     => ~ ( aa_set_real_o(member_real(C),A4)
         => aa_set_real_o(member_real(C),B4) ) ) ).

% DiffE
tff(fact_1806_DiffE,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A4,B4))
     => ~ ( aa_set_o_o(member_o((C)),A4)
         => aa_set_o_o(member_o((C)),B4) ) ) ).

% DiffE
tff(fact_1807_DiffE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A4,B4))
     => ~ ( aa_set_int_o(member_int(C),A4)
         => aa_set_int_o(member_int(C),B4) ) ) ).

% DiffE
tff(fact_1808_DiffE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
     => ~ ( aa_set_nat_o(member_nat(C),A4)
         => aa_set_nat_o(member_nat(C),B4) ) ) ).

% DiffE
tff(fact_1809_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_1810_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D)) ) ) ).

% diff_mono
tff(fact_1811_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_1812_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_left_mono
tff(fact_1813_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_left_mono
tff(fact_1814_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_1815_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_right_mono
tff(fact_1816_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_right_mono
tff(fact_1817_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_1818_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1819_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1820_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1821_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_strict_right_mono
tff(fact_1822_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_strict_right_mono
tff(fact_1823_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_1824_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_strict_left_mono
tff(fact_1825_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_strict_left_mono
tff(fact_1826_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_1827_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1828_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1829_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1830_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_1831_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D)) ) ) ).

% diff_strict_mono
tff(fact_1832_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_1833_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(minus_minus_real(A4),B) = aa_real_real(plus_plus_real(K2),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1834_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1835_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1836_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C )
    <=> ( A = aa_real_real(plus_plus_real(C),B) ) ) ).

% diff_eq_eq
tff(fact_1837_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_1838_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_1839_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1840_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1841_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1842_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),C) ).

% add_diff_eq
tff(fact_1843_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).

% add_diff_eq
tff(fact_1844_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_1845_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_diff_eq2
tff(fact_1846_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_diff_eq2
tff(fact_1847_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_1848_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_add_eq
tff(fact_1849_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_add_eq
tff(fact_1850_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_1851_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1852_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1853_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1854_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(C),B) = A )
     => ( C = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_1855_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_1856_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_1857_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_1858_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% diff_diff_eq
tff(fact_1859_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_1860_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_1861_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_1862_Diff__mono,axiom,
    ! [A4: set_nat,C2: set_nat,D2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D2),B4)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),D2)) ) ) ).

% Diff_mono
tff(fact_1863_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),A4) ).

% Diff_subset
tff(fact_1864_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A4)) = A4 ) ) ) ).

% double_diff
tff(fact_1865_insert__Diff__if,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X3,A4)),B4) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4),insert8211810215607154385at_nat(X3,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))) ).

% insert_Diff_if
tff(fact_1866_insert__Diff__if,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X3,A4)),B4) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),B4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4),insert9069300056098147895at_nat(X3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4))) ).

% insert_Diff_if
tff(fact_1867_insert__Diff__if,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      minus_811609699411566653omplex(insert_complex(X3,A4),B4) = $ite(aa_set_complex_o(member_complex(X3),B4),minus_811609699411566653omplex(A4,B4),insert_complex(X3,minus_811609699411566653omplex(A4,B4))) ).

% insert_Diff_if
tff(fact_1868_insert__Diff__if,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      minus_minus_set_real(insert_real(X3,A4),B4) = $ite(aa_set_real_o(member_real(X3),B4),minus_minus_set_real(A4,B4),insert_real(X3,minus_minus_set_real(A4,B4))) ).

% insert_Diff_if
tff(fact_1869_insert__Diff__if,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      minus_minus_set_o(insert_o((X3),A4),B4) = $ite(aa_set_o_o(member_o((X3)),B4),minus_minus_set_o(A4,B4),insert_o((X3),minus_minus_set_o(A4,B4))) ).

% insert_Diff_if
tff(fact_1870_insert__Diff__if,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      minus_minus_set_int(insert_int(X3,A4),B4) = $ite(aa_set_int_o(member_int(X3),B4),minus_minus_set_int(A4,B4),insert_int(X3,minus_minus_set_int(A4,B4))) ).

% insert_Diff_if
tff(fact_1871_insert__Diff__if,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X3,A4)),B4) = $ite(aa_set_nat_o(member_nat(X3),B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4),insert_nat(X3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ).

% insert_Diff_if
tff(fact_1872_Un__Diff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),C2)) ).

% Un_Diff
tff(fact_1873_Un__Diff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),C2)) ).

% Un_Diff
tff(fact_1874_psubset__imp__ex__mem,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A4),B4)
     => ? [B3: complex] : aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1875_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A4),B4)
     => ? [B3: real] : aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1876_psubset__imp__ex__mem,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),B4)
     => ? [B3: $o] : aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1877_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ? [B3: int] : aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1878_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
     => ? [B3: nat] : aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1879_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_le_eq
tff(fact_1880_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_1881_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_1882_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% le_diff_eq
tff(fact_1883_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_1884_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_1885_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_1886_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_1887_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1888_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1889_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1890_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1891_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1892_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1893_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1894_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1895_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% less_diff_eq
tff(fact_1896_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_1897_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_1898_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_less_eq
tff(fact_1899_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_1900_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_1901_Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% Diff_insert
tff(fact_1902_Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_1903_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : minus_minus_set_o(A4,insert_o((A),B4)) = minus_minus_set_o(minus_minus_set_o(A4,B4),insert_o((A),bot_bot_set_o)) ).

% Diff_insert
tff(fact_1904_Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] : minus_minus_set_int(A4,insert_int(A,B4)) = minus_minus_set_int(minus_minus_set_int(A4,B4),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_1905_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_1906_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_1907_insert__Diff,axiom,
    ! [A: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(A),A4)
     => ( insert_complex(A,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = A4 ) ) ).

% insert_Diff
tff(fact_1908_insert__Diff,axiom,
    ! [A: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ( insert_real(A,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = A4 ) ) ).

% insert_Diff
tff(fact_1909_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_1910_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( insert_o((A),minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = A4 ) ) ).

% insert_Diff
tff(fact_1911_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( insert_int(A,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = A4 ) ) ).

% insert_Diff
tff(fact_1912_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_1913_Diff__insert2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))),B4) ).

% Diff_insert2
tff(fact_1914_Diff__insert2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))),B4) ).

% Diff_insert2
tff(fact_1915_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : minus_minus_set_o(A4,insert_o((A),B4)) = minus_minus_set_o(minus_minus_set_o(A4,insert_o((A),bot_bot_set_o)),B4) ).

% Diff_insert2
tff(fact_1916_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B4: set_int] : minus_minus_set_int(A4,insert_int(A,B4)) = minus_minus_set_int(minus_minus_set_int(A4,insert_int(A,bot_bot_set_int)),B4) ).

% Diff_insert2
tff(fact_1917_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))),B4) ).

% Diff_insert2
tff(fact_1918_Diff__insert__absorb,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X3,A4)),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1919_Diff__insert__absorb,axiom,
    ! [X3: complex,A4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(X3),A4)
     => ( minus_811609699411566653omplex(insert_complex(X3,A4),insert_complex(X3,bot_bot_set_complex)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1920_Diff__insert__absorb,axiom,
    ! [X3: real,A4: set_real] :
      ( ~ aa_set_real_o(member_real(X3),A4)
     => ( minus_minus_set_real(insert_real(X3,A4),insert_real(X3,bot_bot_set_real)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1921_Diff__insert__absorb,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X3,A4)),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1922_Diff__insert__absorb,axiom,
    ! [X3: $o,A4: set_o] :
      ( ~ aa_set_o_o(member_o((X3)),A4)
     => ( minus_minus_set_o(insert_o((X3),A4),insert_o((X3),bot_bot_set_o)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1923_Diff__insert__absorb,axiom,
    ! [X3: int,A4: set_int] :
      ( ~ aa_set_int_o(member_int(X3),A4)
     => ( minus_minus_set_int(insert_int(X3,A4),insert_int(X3,bot_bot_set_int)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1924_Diff__insert__absorb,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X3),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X3,A4)),insert_nat(X3,bot_bot_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1925_subset__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),insert8211810215607154385at_nat(X3,C2)))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),C2))
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4) ) ) ).

% subset_Diff_insert
tff(fact_1926_subset__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),insert9069300056098147895at_nat(X3,C2)))
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),C2))
        & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4) ) ) ).

% subset_Diff_insert
tff(fact_1927_subset__Diff__insert,axiom,
    ! [A4: set_complex,B4: set_complex,X3: complex,C2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),minus_811609699411566653omplex(B4,insert_complex(X3,C2)))
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(A4),minus_811609699411566653omplex(B4,C2))
        & ~ aa_set_complex_o(member_complex(X3),A4) ) ) ).

% subset_Diff_insert
tff(fact_1928_subset__Diff__insert,axiom,
    ! [A4: set_real,B4: set_real,X3: real,C2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),minus_minus_set_real(B4,insert_real(X3,C2)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A4),minus_minus_set_real(B4,C2))
        & ~ aa_set_real_o(member_real(X3),A4) ) ) ).

% subset_Diff_insert
tff(fact_1929_subset__Diff__insert,axiom,
    ! [A4: set_o,B4: set_o,X3: $o,C2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),minus_minus_set_o(B4,insert_o((X3),C2)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A4),minus_minus_set_o(B4,C2))
        & ~ aa_set_o_o(member_o((X3)),A4) ) ) ).

% subset_Diff_insert
tff(fact_1930_subset__Diff__insert,axiom,
    ! [A4: set_int,B4: set_int,X3: int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),minus_minus_set_int(B4,insert_int(X3,C2)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),minus_minus_set_int(B4,C2))
        & ~ aa_set_int_o(member_int(X3),A4) ) ) ).

% subset_Diff_insert
tff(fact_1931_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),insert_nat(X3,C2)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),C2))
        & ~ aa_set_nat_o(member_nat(X3),A4) ) ) ).

% subset_Diff_insert
tff(fact_1932_Diff__subset__conv,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),C2)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) ) ).

% Diff_subset_conv
tff(fact_1933_Diff__subset__conv,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),C2)
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) ) ).

% Diff_subset_conv
tff(fact_1934_Diff__partition,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_1935_Diff__partition,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_1936_subset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X3,B4))
    <=> $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))),B4),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1937_subset__insert__iff,axiom,
    ! [A4: set_complex,X3: complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),insert_complex(X3,B4))
    <=> $ite(aa_set_complex_o(member_complex(X3),A4),aa_set_complex_o(ord_le211207098394363844omplex(minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex))),B4),aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1938_subset__insert__iff,axiom,
    ! [A4: set_real,X3: real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),insert_real(X3,B4))
    <=> $ite(aa_set_real_o(member_real(X3),A4),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real))),B4),aa_set_real_o(ord_less_eq_set_real(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1939_subset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X3,B4))
    <=> $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))),B4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)) ) ).

% subset_insert_iff
tff(fact_1940_subset__insert__iff,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X3),B4))
    <=> $ite(aa_set_o_o(member_o((X3)),A4),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o))),B4),aa_set_o_o(ord_less_eq_set_o(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1941_subset__insert__iff,axiom,
    ! [A4: set_int,X3: int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),insert_int(X3,B4))
    <=> $ite(aa_set_int_o(member_int(X3),A4),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int))),B4),aa_set_int_o(ord_less_eq_set_int(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1942_subset__insert__iff,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(X3,B4))
    <=> $ite(aa_set_nat_o(member_nat(X3),A4),aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat))),B4),aa_set_nat_o(ord_less_eq_set_nat(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1943_Diff__single__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))),B4)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X3,B4)) ) ).

% Diff_single_insert
tff(fact_1944_Diff__single__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))),B4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X3,B4)) ) ).

% Diff_single_insert
tff(fact_1945_Diff__single__insert,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o))),B4)
     => aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X3),B4)) ) ).

% Diff_single_insert
tff(fact_1946_Diff__single__insert,axiom,
    ! [A4: set_int,X3: int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int))),B4)
     => aa_set_int_o(ord_less_eq_set_int(A4),insert_int(X3,B4)) ) ).

% Diff_single_insert
tff(fact_1947_Diff__single__insert,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat))),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A4),insert_nat(X3,B4)) ) ).

% Diff_single_insert
tff(fact_1948_power2__commute,axiom,
    ! [X3: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1949_power2__commute,axiom,
    ! [X3: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Y),X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1950_power2__commute,axiom,
    ! [X3: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1951_power2__commute,axiom,
    ! [X3: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1952_psubset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A4),insert9069300056098147895at_nat(X3,B4))
    <=> $ite(
          aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),B4),
          aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A4),B4),
          $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),A4),aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))),B4),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1953_psubset__insert__iff,axiom,
    ! [A4: set_complex,X3: complex,B4: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A4),insert_complex(X3,B4))
    <=> $ite(
          aa_set_complex_o(member_complex(X3),B4),
          aa_set_complex_o(ord_less_set_complex(A4),B4),
          $ite(aa_set_complex_o(member_complex(X3),A4),aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex))),B4),aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1954_psubset__insert__iff,axiom,
    ! [A4: set_real,X3: real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A4),insert_real(X3,B4))
    <=> $ite(
          aa_set_real_o(member_real(X3),B4),
          aa_set_real_o(ord_less_set_real(A4),B4),
          $ite(aa_set_real_o(member_real(X3),A4),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real))),B4),aa_set_real_o(ord_less_eq_set_real(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1955_psubset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),insert8211810215607154385at_nat(X3,B4))
    <=> $ite(
          aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B4),
          aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),B4),
          $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))),B4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1956_psubset__insert__iff,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),insert_o((X3),B4))
    <=> $ite(
          aa_set_o_o(member_o((X3)),B4),
          aa_set_o_o(ord_less_set_o(A4),B4),
          $ite(aa_set_o_o(member_o((X3)),A4),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o))),B4),aa_set_o_o(ord_less_eq_set_o(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1957_psubset__insert__iff,axiom,
    ! [A4: set_int,X3: int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),insert_int(X3,B4))
    <=> $ite(
          aa_set_int_o(member_int(X3),B4),
          aa_set_int_o(ord_less_set_int(A4),B4),
          $ite(aa_set_int_o(member_int(X3),A4),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int))),B4),aa_set_int_o(ord_less_eq_set_int(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1958_psubset__insert__iff,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),insert_nat(X3,B4))
    <=> $ite(
          aa_set_nat_o(member_nat(X3),B4),
          aa_set_nat_o(ord_less_set_nat(A4),B4),
          $ite(aa_set_nat_o(member_nat(X3),A4),aa_set_nat_o(ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat))),B4),aa_set_nat_o(ord_less_eq_set_nat(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1959_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y)) ).

% inf_sup_ord(4)
tff(fact_1960_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)) ).

% inf_sup_ord(4)
tff(fact_1961_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)) ).

% inf_sup_ord(4)
tff(fact_1962_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)) ).

% inf_sup_ord(4)
tff(fact_1963_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)) ).

% inf_sup_ord(4)
tff(fact_1964_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y)) ).

% inf_sup_ord(3)
tff(fact_1965_inf__sup__ord_I3_J,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)) ).

% inf_sup_ord(3)
tff(fact_1966_inf__sup__ord_I3_J,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)) ).

% inf_sup_ord(3)
tff(fact_1967_inf__sup__ord_I3_J,axiom,
    ! [X3: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)) ).

% inf_sup_ord(3)
tff(fact_1968_inf__sup__ord_I3_J,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)) ).

% inf_sup_ord(3)
tff(fact_1969_le__supE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X3)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X3)
         => ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X3) ) ) ).

% le_supE
tff(fact_1970_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X3)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A),X3)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B),X3) ) ) ).

% le_supE
tff(fact_1971_le__supE,axiom,
    ! [A: rat,B: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X3)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X3) ) ) ).

% le_supE
tff(fact_1972_le__supE,axiom,
    ! [A: nat,B: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X3)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X3) ) ) ).

% le_supE
tff(fact_1973_le__supE,axiom,
    ! [A: int,B: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X3)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X3) ) ) ).

% le_supE
tff(fact_1974_le__supI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X3)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X3) ) ) ).

% le_supI
tff(fact_1975_le__supI,axiom,
    ! [A: set_nat,X3: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),X3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),X3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X3) ) ) ).

% le_supI
tff(fact_1976_le__supI,axiom,
    ! [A: rat,X3: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X3) ) ) ).

% le_supI
tff(fact_1977_le__supI,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X3) ) ) ).

% le_supI
tff(fact_1978_le__supI,axiom,
    ! [A: int,X3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X3) ) ) ).

% le_supI
tff(fact_1979_sup__ge1,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y)) ).

% sup_ge1
tff(fact_1980_sup__ge1,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)) ).

% sup_ge1
tff(fact_1981_sup__ge1,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)) ).

% sup_ge1
tff(fact_1982_sup__ge1,axiom,
    ! [X3: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)) ).

% sup_ge1
tff(fact_1983_sup__ge1,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)) ).

% sup_ge1
tff(fact_1984_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y)) ).

% sup_ge2
tff(fact_1985_sup__ge2,axiom,
    ! [Y: set_nat,X3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)) ).

% sup_ge2
tff(fact_1986_sup__ge2,axiom,
    ! [Y: rat,X3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)) ).

% sup_ge2
tff(fact_1987_sup__ge2,axiom,
    ! [Y: nat,X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)) ).

% sup_ge2
tff(fact_1988_sup__ge2,axiom,
    ! [Y: int,X3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)) ).

% sup_ge2
tff(fact_1989_le__supI1,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),A)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI1
tff(fact_1990_le__supI1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),A)
     => aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI1
tff(fact_1991_le__supI1,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI1
tff(fact_1992_le__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI1
tff(fact_1993_le__supI1,axiom,
    ! [X3: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI1
tff(fact_1994_le__supI2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),B)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI2
tff(fact_1995_le__supI2,axiom,
    ! [X3: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),B)
     => aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI2
tff(fact_1996_le__supI2,axiom,
    ! [X3: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI2
tff(fact_1997_le__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI2
tff(fact_1998_le__supI2,axiom,
    ! [X3: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI2
tff(fact_1999_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(D),B)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_2000_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D),B)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_2001_sup_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ) ).

% sup.mono
tff(fact_2002_sup_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_2003_sup_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_2004_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),C)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),D)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D)) ) ) ).

% sup_mono
tff(fact_2005_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),D)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D)) ) ) ).

% sup_mono
tff(fact_2006_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D)) ) ) ).

% sup_mono
tff(fact_2007_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D)) ) ) ).

% sup_mono
tff(fact_2008_sup__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D)) ) ) ).

% sup_mono
tff(fact_2009_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z),X3)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X3) ) ) ).

% sup_least
tff(fact_2010_sup__least,axiom,
    ! [Y: set_nat,X3: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Z),X3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X3) ) ) ).

% sup_least
tff(fact_2011_sup__least,axiom,
    ! [Y: rat,X3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)),X3) ) ) ).

% sup_least
tff(fact_2012_sup__least,axiom,
    ! [Y: nat,X3: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X3) ) ) ).

% sup_least
tff(fact_2013_sup__least,axiom,
    ! [Y: int,X3: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X3) ) ) ).

% sup_least
tff(fact_2014_le__iff__sup,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),Y)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2015_le__iff__sup,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2016_le__iff__sup,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2017_le__iff__sup,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2018_le__iff__sup,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2019_sup_OorderE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_2020_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_2021_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_2022_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_2023_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_2024_sup_OorderI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A) ) ).

% sup.orderI
tff(fact_2025_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% sup.orderI
tff(fact_2026_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% sup.orderI
tff(fact_2027_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% sup.orderI
tff(fact_2028_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% sup.orderI
tff(fact_2029_sup__unique,axiom,
    ! [F: fun_se177437144654672861at_nat,X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X5),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X5),Y3))
     => ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X5),Y3))
       => ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y3),X5)
             => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z2),X5)
               => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,Y3),Z2)),X5) ) )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_2030_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X3: set_nat,Y: set_nat] :
      ( ! [X5: set_nat,Y3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3))
     => ( ! [X5: set_nat,Y3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Y3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3))
       => ( ! [X5: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(Y3),X5)
             => ( aa_set_nat_o(ord_less_eq_set_nat(Z2),X5)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),Z2)),X5) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_2031_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X3: rat,Y: rat] :
      ( ! [X5: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3))
     => ( ! [X5: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3))
       => ( ! [X5: rat,Y3: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),X5)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),Z2)),X5) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_2032_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X3: nat,Y: nat] :
      ( ! [X5: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3))
     => ( ! [X5: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3))
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X5)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),Z2)),X5) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_2033_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X3: int,Y: int] :
      ( ! [X5: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),aa_int_int(aa_int_fun_int_int(F,X5),Y3))
     => ( ! [X5: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),aa_int_int(aa_int_fun_int_int(F,X5),Y3))
       => ( ! [X5: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),X5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y3),Z2)),X5) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = aa_int_int(aa_int_fun_int_int(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_2034_sup_Oabsorb1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2035_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2036_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2037_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2038_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2039_sup_Oabsorb2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2040_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2041_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2042_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2043_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2044_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_2045_sup__absorb1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_2046_sup__absorb1,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_2047_sup__absorb1,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_2048_sup__absorb1,axiom,
    ! [Y: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_2049_sup__absorb2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),Y)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2050_sup__absorb2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2051_sup__absorb2,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2052_sup__absorb2,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2053_sup__absorb2,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2054_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
         => ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A) ) ) ).

% sup.boundedE
tff(fact_2055_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% sup.boundedE
tff(fact_2056_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% sup.boundedE
tff(fact_2057_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% sup.boundedE
tff(fact_2058_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% sup.boundedE
tff(fact_2059_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_2060_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_2061_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_2062_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_2063_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_2064_sup_Oorder__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
    <=> ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_2065_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_2066_sup_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_2067_sup_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_2068_sup_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_2069_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded1
tff(fact_2070_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded1
tff(fact_2071_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded1
tff(fact_2072_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded1
tff(fact_2073_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded1
tff(fact_2074_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded2
tff(fact_2075_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded2
tff(fact_2076_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded2
tff(fact_2077_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded2
tff(fact_2078_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded2
tff(fact_2079_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2080_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2081_sup_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2082_sup_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2083_sup_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2084_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2085_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2086_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2087_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2088_sup_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2089_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_2090_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
     => aa_set_nat_o(ord_less_eq_set_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_2091_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI1
tff(fact_2092_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_2093_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_2094_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),B)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_2095_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
     => aa_set_nat_o(ord_less_eq_set_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_2096_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI2
tff(fact_2097_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_2098_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_2099_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X5),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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))),Mb))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X5: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X5: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),Nb) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X5)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_2100_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X5),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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))),Mb))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X5: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X5),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X5: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),Nb) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X5)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_2101_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),X3)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) ) ).

% in_children_def
tff(fact_2102_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X3: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Mia),Maa)),Dega,TreeLista,Summarya)),X3) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_2103_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( X3 = Mia )
          | ( X3 = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_2104_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),X3)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( X3 = Mia )
          | ( X3 = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_2105_pred__list__to__short,axiom,
    ! [Dega: nat,X3: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_2106_succ__list__to__short,axiom,
    ! [Dega: nat,Mia: nat,X3: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),X3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_2107_both__member__options__ding,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),X3) ) ) ) ).

% both_member_options_ding
tff(fact_2108_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2109_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2110_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2111_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2112_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2113_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2114_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2115_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2116_low__inv,axiom,
    ! [X3: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),X3)),Nb) = X3 ) ) ).

% low_inv
tff(fact_2117_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K2: 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,Mb)),Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K2) ).

% Suc_diff_diff
tff(fact_2118_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_2119_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_2120_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: 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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ).

% diff_diff_left
tff(fact_2121_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_2122_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_2123_high__def,axiom,
    ! [X3: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_2124_high__inv,axiom,
    ! [X3: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),X3)),Nb) = Y ) ) ).

% high_inv
tff(fact_2125_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_2126_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_2127_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_2128_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_2129_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_2130_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2131_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2132_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2133_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2134_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2135_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2136_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2137_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_2138_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_2139_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_2140_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2141_mult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% mult_1
tff(fact_2142_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_2143_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_2144_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_2145_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_2146_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_2147_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),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)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_2148_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_2149_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_2150_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2151_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2152_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V2: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V2))) ).

% distrib_right_numeral
tff(fact_2153_distrib__right__numeral,axiom,
    ! [A: real,B: real,V2: num] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,V2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,V2))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V2))) ).

% distrib_right_numeral
tff(fact_2154_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V2)) = 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,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V2))) ).

% distrib_right_numeral
tff(fact_2155_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V2)) = 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,V2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V2))) ).

% distrib_right_numeral
tff(fact_2156_distrib__right__numeral,axiom,
    ! [A: int,B: int,V2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V2)) = 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,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V2))) ).

% distrib_right_numeral
tff(fact_2157_distrib__left__numeral,axiom,
    ! [V2: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),C)) ).

% distrib_left_numeral
tff(fact_2158_distrib__left__numeral,axiom,
    ! [V2: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),C)) ).

% distrib_left_numeral
tff(fact_2159_distrib__left__numeral,axiom,
    ! [V2: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),C)) ).

% distrib_left_numeral
tff(fact_2160_distrib__left__numeral,axiom,
    ! [V2: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),C)) ).

% distrib_left_numeral
tff(fact_2161_distrib__left__numeral,axiom,
    ! [V2: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),C)) ).

% distrib_left_numeral
tff(fact_2162_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V2: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V2))) ).

% left_diff_distrib_numeral
tff(fact_2163_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V2: num] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),aa_num_real(numeral_numeral_real,V2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,V2))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V2))) ).

% left_diff_distrib_numeral
tff(fact_2164_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V2)) = 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,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V2))) ).

% left_diff_distrib_numeral
tff(fact_2165_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V2)) = 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,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V2))) ).

% left_diff_distrib_numeral
tff(fact_2166_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_2167_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_2168_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_2169_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_2170_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))),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,K2),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2171_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2172_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_2173_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M: nat] :
              ( ( some_nat(M) = vEBT_vebt_mint(Summarya) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_2174_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2175_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2176_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2177_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2178_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2179_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2180_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2181_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2182_power__add__numeral2,axiom,
    ! [A: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2183_power__add__numeral2,axiom,
    ! [A: real,Mb: num,Nb: num,B: real] : aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2184_power__add__numeral2,axiom,
    ! [A: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2185_power__add__numeral2,axiom,
    ! [A: nat,Mb: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2186_power__add__numeral2,axiom,
    ! [A: int,Mb: num,Nb: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_2187_power__add__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_2188_power__add__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_2189_power__add__numeral,axiom,
    ! [A: rat,Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_2190_power__add__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_2191_power__add__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_2192_diff__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% diff_mult_distrib2
tff(fact_2193_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% diff_mult_distrib
tff(fact_2194_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2195_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2196_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2197_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2198_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.assoc
tff(fact_2199_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_2200_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_2201_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_2202_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),B) = aa_real_real(times_times_real(B),A) ).

% mult.commute
tff(fact_2203_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% mult.commute
tff(fact_2204_mult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% mult.commute
tff(fact_2205_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% mult.commute
tff(fact_2206_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(times_times_real(B),aa_real_real(times_times_real(A),C)) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.left_commute
tff(fact_2207_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.left_commute
tff(fact_2208_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_2209_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_2210_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),E)),C) ).

% combine_common_factor
tff(fact_2211_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2212_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2213_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_2214_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% distrib_right
tff(fact_2215_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_2216_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_2217_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_2218_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% distrib_left
tff(fact_2219_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_2220_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_2221_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_2222_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2223_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2224_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2225_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2226_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2227_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2228_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2229_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2230_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2231_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2232_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),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)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_2233_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_2234_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),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)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_2235_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_2236_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = 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)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2237_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_2238_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_divide
tff(fact_2239_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(divide_divide_real(A),B)),Nb) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ).

% power_divide
tff(fact_2240_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_2241_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K2: nat,I: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),I)) ) ) ).

% zero_induct_lemma
tff(fact_2242_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_2243_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2244_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_2245_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_2246_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_2247_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2248_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2249_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_2250_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_2251_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_2252_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_2253_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K2) ) ).

% less_imp_diff_less
tff(fact_2254_power__commuting__commutes,axiom,
    ! [X3: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(X3),Y) = aa_complex_complex(times_times_complex(Y),X3) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X3),Nb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(X3),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_2255_power__commuting__commutes,axiom,
    ! [X3: real,Y: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(X3),Y) = aa_real_real(times_times_real(Y),X3) )
     => ( aa_real_real(times_times_real(aa_nat_real(power_power_real(X3),Nb)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(power_power_real(X3),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_2256_power__commuting__commutes,axiom,
    ! [X3: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X3) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),Nb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X3),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_2257_power__commuting__commutes,axiom,
    ! [X3: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X3) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_2258_power__commuting__commutes,axiom,
    ! [X3: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X3) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X3),Nb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X3),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_2259_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_mult_distrib
tff(fact_2260_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(times_times_real(A),B)),Nb) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ).

% power_mult_distrib
tff(fact_2261_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_2262_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_2263_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ).

% power_mult_distrib
tff(fact_2264_power__commutes,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_commutes
tff(fact_2265_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(power_power_real(A),Nb)),A) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb)) ).

% power_commutes
tff(fact_2266_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_2267_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_commutes
tff(fact_2268_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_commutes
tff(fact_2269_Suc__mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_2270_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_2271_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_2272_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_2273_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_2274_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_2275_le__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_2276_eq__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_2277_power__mult,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),Nb) ).

% power_mult
tff(fact_2278_power__mult,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),Mb)),Nb) ).

% power_mult
tff(fact_2279_power__mult,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Mb)),Nb) ).

% power_mult
tff(fact_2280_power__mult,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Mb)),Nb) ).

% power_mult
tff(fact_2281_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_2282_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_2283_diff__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_cancel2
tff(fact_2284_Nat_Odiff__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_2285_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K2: 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,K2),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ).

% mult_le_mono2
tff(fact_2286_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ).

% mult_le_mono1
tff(fact_2287_mult__le__mono,axiom,
    ! [I: nat,J: nat,K2: 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,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2288_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_2289_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_2290_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K2) ).

% left_add_mult_distrib
tff(fact_2291_add__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% add_mult_distrib2
tff(fact_2292_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% add_mult_distrib
tff(fact_2293_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_2294_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_2295_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2296_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2297_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2298_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D) )
    <=> ( C = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2299_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2300_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2301_square__diff__square__factored,axiom,
    ! [X3: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X3),X3)),aa_real_real(times_times_real(Y),Y)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(X3),Y)),aa_real_real(minus_minus_real(X3),Y)) ).

% square_diff_square_factored
tff(fact_2302_square__diff__square__factored,axiom,
    ! [X3: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)) ).

% square_diff_square_factored
tff(fact_2303_square__diff__square__factored,axiom,
    ! [X3: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)) ).

% square_diff_square_factored
tff(fact_2304_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),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)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_2305_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_2306_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2307_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2308_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2309_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2310_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2311_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2312_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_2313_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_2314_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_2315_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_2316_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_2317_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_2318_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_2319_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_2320_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_2321_square__diff__one__factored,axiom,
    ! [X3: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X3),X3)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X3),one_one_complex)),aa_complex_complex(minus_minus_complex(X3),one_one_complex)) ).

% square_diff_one_factored
tff(fact_2322_square__diff__one__factored,axiom,
    ! [X3: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X3),X3)),one_one_real) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(X3),one_one_real)),aa_real_real(minus_minus_real(X3),one_one_real)) ).

% square_diff_one_factored
tff(fact_2323_square__diff__one__factored,axiom,
    ! [X3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),one_one_rat)) ).

% square_diff_one_factored
tff(fact_2324_square__diff__one__factored,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),one_one_int)) ).

% square_diff_one_factored
tff(fact_2325_power__one__over,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_one_over
tff(fact_2326_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(divide_divide_real(one_one_real),A)),Nb) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(A),Nb)) ).

% power_one_over
tff(fact_2327_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_2328_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_2329_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_2330_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_2331_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_2332_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_2333_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_2334_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_2335_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_2336_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_2337_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_2338_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_2339_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_2340_left__right__inverse__power,axiom,
    ! [X3: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(X3),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X3),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_2341_left__right__inverse__power,axiom,
    ! [X3: real,Y: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(X3),Y) = one_one_real )
     => ( aa_real_real(times_times_real(aa_nat_real(power_power_real(X3),Nb)),aa_nat_real(power_power_real(Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_2342_left__right__inverse__power,axiom,
    ! [X3: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2343_left__right__inverse__power,axiom,
    ! [X3: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2344_left__right__inverse__power,axiom,
    ! [X3: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X3),Nb)),aa_nat_int(power_power_int(Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2345_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_2346_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_Suc
tff(fact_2347_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb)) ).

% power_Suc
tff(fact_2348_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_2349_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_2350_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_2351_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).

% power_Suc2
tff(fact_2352_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),Nb)),A) ).

% power_Suc2
tff(fact_2353_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_2354_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_2355_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_2356_less__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_2357_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_2358_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: 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),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ).

% less_diff_conv
tff(fact_2359_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_2360_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_2361_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: 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) = K2 )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_2362_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_2363_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_2364_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_2365_le__diff__conv,axiom,
    ! [J: nat,K2: 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),K2)),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),K2)) ) ).

% le_diff_conv
tff(fact_2366_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_2367_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb)) ).

% power_add
tff(fact_2368_power__add,axiom,
    ! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_2369_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_2370_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_2371_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_2372_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_2373_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_2374_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2375_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2376_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2377_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2378_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2379_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_2380_power__gt1__lemma,axiom,
    ! [A: rat,Nb: 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),Nb))) ) ).

% power_gt1_lemma
tff(fact_2381_power__gt1__lemma,axiom,
    ! [A: nat,Nb: 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),Nb))) ) ).

% power_gt1_lemma
tff(fact_2382_power__gt1__lemma,axiom,
    ! [A: int,Nb: 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),Nb))) ) ).

% power_gt1_lemma
tff(fact_2383_power__less__power__Suc,axiom,
    ! [A: real,Nb: 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(power_power_real(A),Nb)),aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2384_power__less__power__Suc,axiom,
    ! [A: rat,Nb: 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),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2385_power__less__power__Suc,axiom,
    ! [A: nat,Nb: 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2386_power__less__power__Suc,axiom,
    ! [A: int,Nb: 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),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2387_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),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),K2)) ) ) ).

% less_diff_conv2
tff(fact_2388_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2389_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2390_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2391_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2392_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2393_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_2394_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2_right
tff(fact_2395_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_2396_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_2397_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_2398_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_2399_mult__2,axiom,
    ! [Z: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2
tff(fact_2400_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_2401_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_2402_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_2403_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_2404_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(A),A) ).

% power2_eq_square
tff(fact_2405_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% power2_eq_square
tff(fact_2406_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_2407_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_2408_power4__eq__xxxx,axiom,
    ! [X3: complex] : aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(X3),X3)),X3)),X3) ).

% power4_eq_xxxx
tff(fact_2409_power4__eq__xxxx,axiom,
    ! [X3: real] : aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(X3),X3)),X3)),X3) ).

% power4_eq_xxxx
tff(fact_2410_power4__eq__xxxx,axiom,
    ! [X3: rat] : aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),X3)),X3) ).

% power4_eq_xxxx
tff(fact_2411_power4__eq__xxxx,axiom,
    ! [X3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),X3)),X3)),X3) ).

% power4_eq_xxxx
tff(fact_2412_power4__eq__xxxx,axiom,
    ! [X3: int] : aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),X3)),X3) ).

% power4_eq_xxxx
tff(fact_2413_power__even__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2414_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2415_power__even__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2416_power__even__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2417_diff__le__diff__pow,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))) ) ).

% diff_le_diff_pow
tff(fact_2418_power2__sum,axiom,
    ! [X3: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3)),Y)) ).

% power2_sum
tff(fact_2419_power2__sum,axiom,
    ! [X3: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(plus_plus_real(X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_sum
tff(fact_2420_power2__sum,axiom,
    ! [X3: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_sum
tff(fact_2421_power2__sum,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_sum
tff(fact_2422_power2__sum,axiom,
    ! [X3: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_sum
tff(fact_2423_power2__diff,axiom,
    ! [X3: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3)),Y)) ).

% power2_diff
tff(fact_2424_power2__diff,axiom,
    ! [X3: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_diff
tff(fact_2425_power2__diff,axiom,
    ! [X3: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_diff
tff(fact_2426_power2__diff,axiom,
    ! [X3: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3)),Y)) ).

% power2_diff
tff(fact_2427_add__le__add__imp__diff__le,axiom,
    ! [I: real,K2: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K2)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K2)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2428_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K2: rat,Nb: 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),K2)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K2))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K2))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2429_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,Nb: 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),K2)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2430_add__le__add__imp__diff__le,axiom,
    ! [I: int,K2: int,Nb: 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),K2)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2431_add__le__imp__le__diff,axiom,
    ! [I: real,K2: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I),K2)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(minus_minus_real(Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_2432_add__le__imp__le__diff,axiom,
    ! [I: rat,K2: rat,Nb: 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),K2)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_2433_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_2434_add__le__imp__le__diff,axiom,
    ! [I: int,K2: int,Nb: 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),K2)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_2435_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_real_real(plus_plus_real(B),one_one_real)) ) ).

% add_mono1
tff(fact_2436_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_2437_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_2438_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_2439_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),one_one_real)) ).

% less_add_one
tff(fact_2440_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_2441_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_2442_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_2443_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2444_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2445_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2446_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2447_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_2448_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_2449_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Va: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Mia)
         => ( ( Maa != Mia )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_2450_sum__squares__bound,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)),Y)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2451_sum__squares__bound,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X3)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2452_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_2453_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia != Maa )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_2454_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_2455_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_2456_field__less__half__sum,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2457_field__less__half__sum,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2458_div__nat__eqI,axiom,
    ! [Nb: nat,Q3: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_2459_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_2460_mul__shift,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X3)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_2461_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% semiring_norm(13)
tff(fact_2462_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_2463_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Mb),one) = Mb ).

% semiring_norm(11)
tff(fact_2464_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_2465_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_2466_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_2467_num__double,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_2468_power__mult__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_2469_power__mult__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_2470_power__mult__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_2471_power__mult__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_2472_diff__commute,axiom,
    ! [I: nat,J: nat,K2: 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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K2)),J) ).

% diff_commute
tff(fact_2473_div__mult2__numeral__eq,axiom,
    ! [A: nat,K2: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K2),L))) ).

% div_mult2_numeral_eq
tff(fact_2474_div__mult2__numeral__eq,axiom,
    ! [A: int,K2: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K2),L))) ).

% div_mult2_numeral_eq
tff(fact_2475_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).

% option.sel
tff(fact_2476_option_Osel,axiom,
    ! [X2: nat] : the_nat(some_nat(X2)) = X2 ).

% option.sel
tff(fact_2477_option_Osel,axiom,
    ! [X2: num] : the_num(some_num(X2)) = X2 ).

% option.sel
tff(fact_2478_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_2479_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_2480_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_2481_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(times_times_real(A),C))),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_2482_four__x__squared,axiom,
    ! [X3: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_2483_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_2484_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_2485_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_2486_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_2487_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) = aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_2488_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D)) ).

% add_diff_add
tff(fact_2489_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_2490_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K2)) ) ).

% div_le_mono
tff(fact_2491_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_2492_mult__diff__mult,axiom,
    ! [X3: real,Y: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X3),Y)),aa_real_real(times_times_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),aa_real_real(minus_minus_real(Y),B))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(X3),A)),B)) ).

% mult_diff_mult
tff(fact_2493_mult__diff__mult,axiom,
    ! [X3: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),A)),B)) ).

% mult_diff_mult
tff(fact_2494_mult__diff__mult,axiom,
    ! [X3: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),A)),B)) ).

% mult_diff_mult
tff(fact_2495_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_2496_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_2497_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_2498_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_2499_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_2500_field__sum__of__halves,axiom,
    ! [X3: real] : aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X3 ).

% field_sum_of_halves
tff(fact_2501_field__sum__of__halves,axiom,
    ! [X3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X3 ).

% field_sum_of_halves
tff(fact_2502_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_2503_del__x__mi__lets__in__not__minNull,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X3 = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_2504_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mia: nat,X3: nat,Maa: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                $ite(X3 = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_2505_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_2506_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_2507_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_2508_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_2509_divmod__step__eq,axiom,
    ! [L: num,Q3: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),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))),Q3)),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))),Q3)),R2)) ).

% divmod_step_eq
tff(fact_2510_divmod__step__eq,axiom,
    ! [L: num,Q3: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),R2)) ).

% divmod_step_eq
tff(fact_2511_divmod__step__eq,axiom,
    ! [L: num,Q3: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R2,aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q3),R2)) ).

% divmod_step_eq
tff(fact_2512_pred__lesseq__max,axiom,
    ! [Dega: nat,X3: nat,Maa: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_2513_pred__less__length__list,axiom,
    ! [Dega: nat,X3: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_2514_succ__less__length__list,axiom,
    ! [Dega: nat,Mia: nat,X3: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),X3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_2515_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_2516_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_2517_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,X3) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ah(Ta),X3)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_2518_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,X3) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai(Ta),X3)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_2519_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(fequal7949547063978738364at_nat(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv2
tff(fact_2520_singleton__conv2,axiom,
    ! [A: complex] : collect_complex(aa_com5808801942499186519plex_o(fequal_complex,A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_2521_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(fequal_set_nat(A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_2522_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_2523_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv2
tff(fact_2524_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_2525_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_2526_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_2527_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(aTP_Lamm_aj(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv
tff(fact_2528_singleton__conv,axiom,
    ! [A: complex] : collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ak,A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv
tff(fact_2529_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aTP_Lamm_al(A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_2530_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_am(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_2531_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_an,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv
tff(fact_2532_singleton__conv,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(aTP_Lamm_ao,(A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv
tff(fact_2533_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ap,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_2534_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_aq,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_2535_del__x__not__mia,axiom,
    ! [Mia: nat,X3: nat,Maa: nat,Dega: nat,H: nat,L: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(
                                    X3 = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                $ite(X3 = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_2536_del__x__not__mi__new__node__nil,axiom,
    ! [Mia: nat,X3: nat,Maa: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Sn: vEBT_VEBT,Summarya: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(
                                    X3 = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_2537_del__x__not__mi,axiom,
    ! [Mia: nat,X3: nat,Maa: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                   => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(
                                    X3 = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,Newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                $ite(X3 = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_2538_del__in__range,axiom,
    ! [Mia: nat,X3: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),X3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
                xn: nat,
                xn:= 
                  $ite(X3 = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),X3),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(X3 = Mia,xn,Mia),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,h),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(
                                      xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Dega,newlist,Summarya) ) ) ),
                      vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_2539_del__x__mi,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat] :
      ( ( ( X3 = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_2540_del__x__mi__lets__in,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X3 = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,Newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_2541_del__x__mi__lets__in__minNull,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X3 = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_2542_del__x__mia,axiom,
    ! [X3: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( X3 = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,h),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(
                                    xn = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Dega,newlist,Summarya) ) ) ),
                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_2543_succ__greatereq__min,axiom,
    ! [Dega: nat,Mia: nat,X3: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),X3)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_2544_minus__set__def,axiom,
    ! [A4: set_real,B4: set_real] : minus_minus_set_real(A4,B4) = collect_real(minus_404528632214381484real_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% minus_set_def
tff(fact_2545_minus__set__def,axiom,
    ! [A4: set_o,B4: set_o] : minus_minus_set_o(A4,B4) = collect_o(minus_minus_fun_o_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ).

% minus_set_def
tff(fact_2546_minus__set__def,axiom,
    ! [A4: set_int,B4: set_int] : minus_minus_set_int(A4,B4) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% minus_set_def
tff(fact_2547_minus__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] : minus_811609699411566653omplex(A4,B4) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% minus_set_def
tff(fact_2548_minus__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = collec3392354462482085612at_nat(minus_2386458686304230003_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% minus_set_def
tff(fact_2549_minus__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4))) ).

% minus_set_def
tff(fact_2550_minus__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ag(A4),aTP_Lamm_ag(B4))) ).

% minus_set_def
tff(fact_2551_minus__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% minus_set_def
tff(fact_2552_set__diff__eq,axiom,
    ! [A4: set_real,B4: set_real] : minus_minus_set_real(A4,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_ar(A4),B4)) ).

% set_diff_eq
tff(fact_2553_set__diff__eq,axiom,
    ! [A4: set_o,B4: set_o] : minus_minus_set_o(A4,B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_as(A4),B4)) ).

% set_diff_eq
tff(fact_2554_set__diff__eq,axiom,
    ! [A4: set_int,B4: set_int] : minus_minus_set_int(A4,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_at(A4),B4)) ).

% set_diff_eq
tff(fact_2555_set__diff__eq,axiom,
    ! [A4: set_complex,B4: set_complex] : minus_811609699411566653omplex(A4,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_au(A4),B4)) ).

% set_diff_eq
tff(fact_2556_set__diff__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_av(A4),B4)) ).

% set_diff_eq
tff(fact_2557_set__diff__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_aw(A4),B4)) ).

% set_diff_eq
tff(fact_2558_set__diff__eq,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ax(A4),B4)) ).

% set_diff_eq
tff(fact_2559_set__diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ay(A4),B4)) ).

% set_diff_eq
tff(fact_2560_real__arch__pow,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X3)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(power_power_real(X3),N)) ) ).

% real_arch_pow
tff(fact_2561_sup__Un__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num,X: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(sup_su1399052437694960234_num_o(aTP_Lamm_az(R),aTP_Lamm_az(S2)),X),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa),sup_su4061117120043295689um_num(R,S2)) ) ).

% sup_Un_eq2
tff(fact_2562_sup__Un__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num,X: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(sup_su8842149202998597428_num_o(aTP_Lamm_ba(R),aTP_Lamm_ba(S2)),X),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa),sup_su2042722026077122175at_num(R,S2)) ) ).

% sup_Un_eq2
tff(fact_2563_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(sup_su8083968640029086462_nat_o(aTP_Lamm_bb(R),aTP_Lamm_bb(S2)),X),Xa)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,R),S2)) ) ).

% sup_Un_eq2
tff(fact_2564_sup__Un__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int,X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(sup_su7393609867835711046_int_o(aTP_Lamm_bc(R),aTP_Lamm_bc(S2)),X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),sup_su6024340866399070445nt_int(R,S2)) ) ).

% sup_Un_eq2
tff(fact_2565_sup__Un__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aTP_Lamm_bd(R),aTP_Lamm_bd(S2)),X),Xa)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Xa)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R),S2)) ) ).

% sup_Un_eq2
tff(fact_2566_sup__Un__eq,axiom,
    ! [R: set_complex,S2: set_complex,X: complex] :
      ( aa_complex_o(sup_su37214000559163671plex_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S2)),X)
    <=> aa_set_complex_o(member_complex(X),sup_sup_set_complex(R,S2)) ) ).

% sup_Un_eq
tff(fact_2567_sup__Un__eq,axiom,
    ! [R: set_real,S2: set_real,X: real] :
      ( aa_real_o(sup_sup_fun_real_o(aTP_Lamm_a(R),aTP_Lamm_a(S2)),X)
    <=> aa_set_real_o(member_real(X),sup_sup_set_real(R,S2)) ) ).

% sup_Un_eq
tff(fact_2568_sup__Un__eq,axiom,
    ! [R: set_o,S2: set_o,X: $o] :
      ( aa_o_o(sup_sup_fun_o_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S2)),(X))
    <=> aa_set_o_o(member_o((X)),sup_sup_set_o(R,S2)) ) ).

% sup_Un_eq
tff(fact_2569_sup__Un__eq,axiom,
    ! [R: set_int,S2: set_int,X: int] :
      ( aa_int_o(sup_sup_fun_int_o(aTP_Lamm_ab(R),aTP_Lamm_ab(S2)),X)
    <=> aa_set_int_o(member_int(X),sup_sup_set_int(R,S2)) ) ).

% sup_Un_eq
tff(fact_2570_sup__Un__eq,axiom,
    ! [R: set_nat,S2: set_nat,X: nat] :
      ( aa_nat_o(sup_sup_fun_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S2)),X)
    <=> aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,R),S2)) ) ).

% sup_Un_eq
tff(fact_2571_sup__Un__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(sup_su2398265404452057824_nat_o(aTP_Lamm_be(R),aTP_Lamm_be(S2)),X)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R),S2)) ) ).

% sup_Un_eq
tff(fact_2572_Un__def,axiom,
    ! [A4: set_real,B4: set_real] : sup_sup_set_real(A4,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_bf(A4),B4)) ).

% Un_def
tff(fact_2573_Un__def,axiom,
    ! [A4: set_o,B4: set_o] : sup_sup_set_o(A4,B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_bg(A4),B4)) ).

% Un_def
tff(fact_2574_Un__def,axiom,
    ! [A4: set_int,B4: set_int] : sup_sup_set_int(A4,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bh(A4),B4)) ).

% Un_def
tff(fact_2575_Un__def,axiom,
    ! [A4: set_complex,B4: set_complex] : sup_sup_set_complex(A4,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_bi(A4),B4)) ).

% Un_def
tff(fact_2576_Un__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_bj(A4),B4)) ).

% Un_def
tff(fact_2577_Un__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : sup_sup_set_set_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_bk(A4),B4)) ).

% Un_def
tff(fact_2578_Un__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : sup_sup_set_list_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_bl(A4),B4)) ).

% Un_def
tff(fact_2579_Un__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bm(A4),B4)) ).

% Un_def
tff(fact_2580_Un__def,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_bn(A4),B4)) ).

% Un_def
tff(fact_2581_sup__set__def,axiom,
    ! [A4: set_real,B4: set_real] : sup_sup_set_real(A4,B4) = collect_real(sup_sup_fun_real_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% sup_set_def
tff(fact_2582_sup__set__def,axiom,
    ! [A4: set_o,B4: set_o] : sup_sup_set_o(A4,B4) = collect_o(sup_sup_fun_o_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ).

% sup_set_def
tff(fact_2583_sup__set__def,axiom,
    ! [A4: set_int,B4: set_int] : sup_sup_set_int(A4,B4) = collect_int(sup_sup_fun_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% sup_set_def
tff(fact_2584_sup__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] : sup_sup_set_complex(A4,B4) = collect_complex(sup_su37214000559163671plex_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% sup_set_def
tff(fact_2585_sup__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) = collec3392354462482085612at_nat(sup_su5606187154157927750_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% sup_set_def
tff(fact_2586_sup__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : sup_sup_set_set_nat(A4,B4) = collect_set_nat(sup_su7503774895437956671_nat_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4))) ).

% sup_set_def
tff(fact_2587_sup__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : sup_sup_set_list_nat(A4,B4) = collect_list_nat(sup_su6906368799827444965_nat_o(aTP_Lamm_ag(A4),aTP_Lamm_ag(B4))) ).

% sup_set_def
tff(fact_2588_sup__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = collect_nat(sup_sup_fun_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% sup_set_def
tff(fact_2589_sup__set__def,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = collec6321179662152712658at_nat(sup_su2398265404452057824_nat_o(aTP_Lamm_be(A4),aTP_Lamm_be(B4))) ).

% sup_set_def
tff(fact_2590_Collect__disj__eq,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] : collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bo(P),Q)) = sup_sup_set_complex(collect_complex(P),collect_complex(Q)) ).

% Collect_disj_eq
tff(fact_2591_Collect__disj__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] : collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bp(P),Q)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q)) ).

% Collect_disj_eq
tff(fact_2592_Collect__disj__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bq(P),Q)) = sup_sup_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_disj_eq
tff(fact_2593_Collect__disj__eq,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_br(P),Q)) = sup_sup_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_disj_eq
tff(fact_2594_Collect__disj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bs(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_disj_eq
tff(fact_2595_Collect__disj__eq,axiom,
    ! [P: fun_Pr3059359313727237556_nat_o,Q: fun_Pr3059359313727237556_nat_o] : collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_bt(P),Q)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(P)),collec6321179662152712658at_nat(Q)) ).

% Collect_disj_eq
tff(fact_2596_pred__subset__eq,axiom,
    ! [R: set_complex,S2: set_complex] :
      ( ord_le7644322562001619171plex_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S2))
    <=> aa_set_complex_o(ord_le211207098394363844omplex(R),S2) ) ).

% pred_subset_eq
tff(fact_2597_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_a(R),aTP_Lamm_a(S2))
    <=> aa_set_real_o(ord_less_eq_set_real(R),S2) ) ).

% pred_subset_eq
tff(fact_2598_pred__subset__eq,axiom,
    ! [R: set_o,S2: set_o] :
      ( ord_less_eq_fun_o_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S2))
    <=> aa_set_o_o(ord_less_eq_set_o(R),S2) ) ).

% pred_subset_eq
tff(fact_2599_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ab(R),aTP_Lamm_ab(S2))
    <=> aa_set_int_o(ord_less_eq_set_int(R),S2) ) ).

% pred_subset_eq
tff(fact_2600_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R),S2) ) ).

% pred_subset_eq
tff(fact_2601_less__eq__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_eq_set_def
tff(fact_2602_less__eq__set__def,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_eq_set_def
tff(fact_2603_less__eq__set__def,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
    <=> ord_less_eq_fun_o_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4)) ) ).

% less_eq_set_def
tff(fact_2604_less__eq__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4)) ) ).

% less_eq_set_def
tff(fact_2605_less__eq__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4)) ) ).

% less_eq_set_def
tff(fact_2606_Collect__subset,axiom,
    ! [A4: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bu(A4),P))),A4) ).

% Collect_subset
tff(fact_2607_Collect__subset,axiom,
    ! [A4: set_o,P: fun_o_o] : aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bv(A4),P))),A4) ).

% Collect_subset
tff(fact_2608_Collect__subset,axiom,
    ! [A4: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bw(A4),P))),A4) ).

% Collect_subset
tff(fact_2609_Collect__subset,axiom,
    ! [A4: set_complex,P: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bx(A4),P))),A4) ).

% Collect_subset
tff(fact_2610_Collect__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_by(A4),P))),A4) ).

% Collect_subset
tff(fact_2611_Collect__subset,axiom,
    ! [A4: set_set_nat,P: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bz(A4),P)),A4) ).

% Collect_subset
tff(fact_2612_Collect__subset,axiom,
    ! [A4: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ca(A4),P)),A4) ).

% Collect_subset
tff(fact_2613_Collect__subset,axiom,
    ! [A4: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(A4),P))),A4) ).

% Collect_subset
tff(fact_2614_bot__empty__eq2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(bot_bo1070391751820557722_nat_o,X),Xa)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Xa)),bot_bo228742789529271731at_nat) ) ).

% bot_empty_eq2
tff(fact_2615_bot__empty__eq2,axiom,
    ! [X: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa),bot_bo9056780473022590049um_num) ) ).

% bot_empty_eq2
tff(fact_2616_bot__empty__eq2,axiom,
    ! [X: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_2617_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_2618_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X),Xa)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_2619_pred__equals__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Xa3: set_Pr1261947904930325089at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X4,Xa3)),R)
        <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X4,Xa3)),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_2620_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
      ( ! [X4: num,Xa3: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X4,Xa3),R)
        <=> member7279096912039735102um_num(product_Pair_num_num(X4,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_2621_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X4: nat,Xa3: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X4,Xa3),R)
        <=> member9148766508732265716at_num(product_Pair_nat_num(X4,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_2622_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Xa3: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3)),R)
        <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3)),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_2623_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_2624_empty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_cc) ).

% empty_def
tff(fact_2625_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_cd) ).

% empty_def
tff(fact_2626_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_ce) ).

% empty_def
tff(fact_2627_empty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_cf) ).

% empty_def
tff(fact_2628_empty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_cg) ).

% empty_def
tff(fact_2629_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_ch) ).

% empty_def
tff(fact_2630_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_ci) ).

% empty_def
tff(fact_2631_insert__Collect,axiom,
    ! [A: int,P: fun_int_o] : insert_int(A,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cj(A),P)) ).

% insert_Collect
tff(fact_2632_insert__Collect,axiom,
    ! [A: $o,P: fun_o_o] : insert_o((A),collect_o(P)) = collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ck((A)),P)) ).

% insert_Collect
tff(fact_2633_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] : insert9069300056098147895at_nat(A,collec6321179662152712658at_nat(P)) = collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_cl(A),P)) ).

% insert_Collect
tff(fact_2634_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_o] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cm(A),P)) ).

% insert_Collect
tff(fact_2635_insert__Collect,axiom,
    ! [A: complex,P: fun_complex_o] : insert_complex(A,collect_complex(P)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cn(A),P)) ).

% insert_Collect
tff(fact_2636_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] : insert8211810215607154385at_nat(A,collec3392354462482085612at_nat(P)) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_co(A),P)) ).

% insert_Collect
tff(fact_2637_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_o] : insert_set_nat(A,collect_set_nat(P)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cp(A),P)) ).

% insert_Collect
tff(fact_2638_insert__Collect,axiom,
    ! [A: list_nat,P: fun_list_nat_o] : insert_list_nat(A,collect_list_nat(P)) = collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_cq(A),P)) ).

% insert_Collect
tff(fact_2639_insert__compr,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B4) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_cr(A),B4)) ).

% insert_compr
tff(fact_2640_insert__compr,axiom,
    ! [A: real,B4: set_real] : insert_real(A,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_cs(A),B4)) ).

% insert_compr
tff(fact_2641_insert__compr,axiom,
    ! [A: $o,B4: set_o] : insert_o((A),B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_ct((A)),B4)) ).

% insert_compr
tff(fact_2642_insert__compr,axiom,
    ! [A: int,B4: set_int] : insert_int(A,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_cu(A),B4)) ).

% insert_compr
tff(fact_2643_insert__compr,axiom,
    ! [A: nat,B4: set_nat] : insert_nat(A,B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_cv(A),B4)) ).

% insert_compr
tff(fact_2644_insert__compr,axiom,
    ! [A: complex,B4: set_complex] : insert_complex(A,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_cw(A),B4)) ).

% insert_compr
tff(fact_2645_insert__compr,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_cx(A),B4)) ).

% insert_compr
tff(fact_2646_insert__compr,axiom,
    ! [A: set_nat,B4: set_set_nat] : insert_set_nat(A,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_cy(A),B4)) ).

% insert_compr
tff(fact_2647_insert__compr,axiom,
    ! [A: list_nat,B4: set_list_nat] : insert_list_nat(A,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_cz(A),B4)) ).

% insert_compr
tff(fact_2648_less__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A4),B4)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_set_def
tff(fact_2649_less__set__def,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A4),B4)
    <=> ord_less_fun_real_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_set_def
tff(fact_2650_less__set__def,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),B4)
    <=> ord_less_fun_o_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4)) ) ).

% less_set_def
tff(fact_2651_less__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
    <=> ord_less_fun_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4)) ) ).

% less_set_def
tff(fact_2652_less__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
    <=> ord_less_fun_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4)) ) ).

% less_set_def
tff(fact_2653_Collect__conv__if2,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
      collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_da(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).

% Collect_conv_if2
tff(fact_2654_Collect__conv__if2,axiom,
    ! [A: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_db(A),P)) = $ite(aa_complex_o(P,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if2
tff(fact_2655_Collect__conv__if2,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dc(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if2
tff(fact_2656_Collect__conv__if2,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dd(A),P)) = $ite(aa_list_nat_o(P,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ).

% Collect_conv_if2
tff(fact_2657_Collect__conv__if2,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_de(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).

% Collect_conv_if2
tff(fact_2658_Collect__conv__if2,axiom,
    ! [A: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_df((A)),P)) = $ite(aa_o_o(P,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if2
tff(fact_2659_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dg(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_2660_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dh(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_2661_Collect__conv__if,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
      collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_di(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).

% Collect_conv_if
tff(fact_2662_Collect__conv__if,axiom,
    ! [A: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dj(A),P)) = $ite(aa_complex_o(P,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if
tff(fact_2663_Collect__conv__if,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dk(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if
tff(fact_2664_Collect__conv__if,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dl(A),P)) = $ite(aa_list_nat_o(P,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ).

% Collect_conv_if
tff(fact_2665_Collect__conv__if,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_dm(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).

% Collect_conv_if
tff(fact_2666_Collect__conv__if,axiom,
    ! [A: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_dn((A)),P)) = $ite(aa_o_o(P,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if
tff(fact_2667_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_do(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_2668_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dp(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_2669_insert__def,axiom,
    ! [A: int,B4: set_int] : insert_int(A,B4) = sup_sup_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_aq,A)),B4) ).

% insert_def
tff(fact_2670_insert__def,axiom,
    ! [A: $o,B4: set_o] : insert_o((A),B4) = sup_sup_set_o(collect_o(aa_o_fun_o_o(aTP_Lamm_ao,(A))),B4) ).

% insert_def
tff(fact_2671_insert__def,axiom,
    ! [A: complex,B4: set_complex] : insert_complex(A,B4) = sup_sup_set_complex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ak,A)),B4) ).

% insert_def
tff(fact_2672_insert__def,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_an,A))),B4) ).

% insert_def
tff(fact_2673_insert__def,axiom,
    ! [A: set_nat,B4: set_set_nat] : insert_set_nat(A,B4) = sup_sup_set_set_nat(collect_set_nat(aTP_Lamm_al(A)),B4) ).

% insert_def
tff(fact_2674_insert__def,axiom,
    ! [A: list_nat,B4: set_list_nat] : insert_list_nat(A,B4) = sup_sup_set_list_nat(collect_list_nat(aTP_Lamm_am(A)),B4) ).

% insert_def
tff(fact_2675_insert__def,axiom,
    ! [A: nat,B4: set_nat] : insert_nat(A,B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ap,A))),B4) ).

% insert_def
tff(fact_2676_insert__def,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(aTP_Lamm_aj(A))),B4) ).

% insert_def
tff(fact_2677_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ord_le5606610691101069006_nat_o(aTP_Lamm_bd(R),aTP_Lamm_bd(S2))
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(R),S2) ) ).

% pred_subset_eq2
tff(fact_2678_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_az(R),aTP_Lamm_az(S2))
    <=> ord_le880128212290418581um_num(R,S2) ) ).

% pred_subset_eq2
tff(fact_2679_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_ba(R),aTP_Lamm_ba(S2))
    <=> ord_le8085105155179020875at_num(R,S2) ) ).

% pred_subset_eq2
tff(fact_2680_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_bb(R),aTP_Lamm_bb(S2))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,R),S2) ) ).

% pred_subset_eq2
tff(fact_2681_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_bc(R),aTP_Lamm_bc(S2))
    <=> ord_le2843351958646193337nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_2682_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_2683_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_2684_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(m2),m2) ) ).

% numeral_code(2)
tff(fact_2685_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2) ) ).

% numeral_code(2)
tff(fact_2686_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_2687_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_2688_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_2689_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(w),w) ) ).

% power_numeral_even
tff(fact_2690_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_2691_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_2692_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_2693_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V2: nat,TreeLista: list_VEBT_VEBT,Sb: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeLista,Sb)),X3)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_2694_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeLista,Vd)),X3)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_2695_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V2: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V2),TreeLista,Vc)),X3)
    <=> ( ( X3 = Mia )
        | ( X3 = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_2696_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),X3) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia),
        some_nat(Mia),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_2697_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),X3) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),X3),
        some_nat(Maa),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3),some_nat(Mia),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_2698_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),X3),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),X4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X4) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_2699_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),X3),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),X3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_2700_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_2701_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_2702_vebt__delete_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),X3) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),X3) ),
        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),
        $ite(
          ( ( X3 = Mia )
          & ( X3 = Maa ) ),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),
          $let(
            xn: nat,
            xn:= 
              $ite(X3 = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),X3),
            $let(
              minn: nat,
              minn:= 
                $ite(X3 = Mia,xn,Mia),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(TreeLista)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summarya,h),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                $ite(
                                  xn = Maa,
                                  $let(
                                    maxs: option_nat,
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                  Maa ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,sn) ),
                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                              $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,Summarya) ) ) ),
                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_2703_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),X3)
    <=> $ite(
          X3 = Mia,
          $true,
          $ite(
            X3 = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),X3),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_2704_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_2705_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_2706_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_2707_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_2708_insert__simp__norm,axiom,
    ! [X3: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( X3 != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_2709_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,X3: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( X3 != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_2710_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: 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,X3),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_2711_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X3: $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,(X3))),I)
      <=> (X3) ) ) ).

% nth_list_update_eq
tff(fact_2712_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X3: 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,X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_2713_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X3: 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,X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_2714_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_2715_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X3: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(X3)) = Xs ) ) ).

% list_update_beyond
tff(fact_2716_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_2717_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I)
     => ( list_update_int(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_2718_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X3)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_2719_length__list__update,axiom,
    ! [Xs: list_o,I: nat,X3: $o] : size_size_list_o(list_update_o(Xs,I,(X3))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_2720_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] : size_size_list_nat(list_update_nat(Xs,I,X3)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_2721_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X3: int] : size_size_list_int(list_update_int(Xs,I,X3)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_2722_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X3),Xa2)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2)
           => ~ ( ( Xa2 = Mi )
                | ( Xa2 = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2)
             => ~ ( ( Xa2 = Mi )
                  | ( Xa2 = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_2723_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_2724_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_2725_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_2726_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_2727_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2728_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2729_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_2730_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_2731_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_2732_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_2733_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_2734_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_2735_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_2736_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_2737_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_2738_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2739_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2740_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2741_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_2742_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2743_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2744_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2745_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_2746_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2747_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2748_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2749_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_2750_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2751_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2752_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2753_add__eq__0__iff__both__eq__0,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = zero_zero_nat )
    <=> ( ( X3 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2754_zero__eq__add__iff__both__eq__0,axiom,
    ! [X3: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) )
    <=> ( ( X3 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2755_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_2756_add__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add_0
tff(fact_2757_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_2758_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_2759_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_2760_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2761_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_2762_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_2763_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_2764_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_2765_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_2766_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_2767_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_2768_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_2769_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_2770_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_2771_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_2772_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_2773_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_2774_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_2775_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_2776_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_2777_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_2778_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_2779_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_2780_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2781_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_2782_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_2783_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_2784_mult__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2785_mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2786_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2787_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2788_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb1
tff(fact_2789_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_2790_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_2791_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_2792_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb2
tff(fact_2793_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_2794_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_2795_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_2796_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.bounded_iff
tff(fact_2797_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.bounded_iff
tff(fact_2798_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.bounded_iff
tff(fact_2799_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.bounded_iff
tff(fact_2800_max__bot,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(ord_ma7524802468073614006at_nat,bot_bo2099793752762293965at_nat),X3) = X3 ).

% max_bot
tff(fact_2801_max__bot,axiom,
    ! [X3: set_o] : ord_max_set_o(bot_bot_set_o,X3) = X3 ).

% max_bot
tff(fact_2802_max__bot,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),X3) = X3 ).

% max_bot
tff(fact_2803_max__bot,axiom,
    ! [X3: set_int] : ord_max_set_int(bot_bot_set_int,X3) = X3 ).

% max_bot
tff(fact_2804_max__bot,axiom,
    ! [X3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X3) = X3 ).

% max_bot
tff(fact_2805_max__bot2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(ord_ma7524802468073614006at_nat,X3),bot_bo2099793752762293965at_nat) = X3 ).

% max_bot2
tff(fact_2806_max__bot2,axiom,
    ! [X3: set_o] : ord_max_set_o(X3,bot_bot_set_o) = X3 ).

% max_bot2
tff(fact_2807_max__bot2,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X3),bot_bot_set_nat) = X3 ).

% max_bot2
tff(fact_2808_max__bot2,axiom,
    ! [X3: set_int] : ord_max_set_int(X3,bot_bot_set_int) = X3 ).

% max_bot2
tff(fact_2809_max__bot2,axiom,
    ! [X3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),bot_bot_nat) = X3 ).

% max_bot2
tff(fact_2810_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_2811_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_2812_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_2813_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_2814_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_2815_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_2816_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_2817_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2818_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2819_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2820_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2821_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2822_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2823_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2824_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2825_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_2826_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2827_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2828_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2829_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_2830_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2831_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2832_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2833_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(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_2834_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_2835_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_2836_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(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_2837_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_2838_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_2839_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(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_2840_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_2841_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_2842_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(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_2843_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_2844_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_2845_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_2846_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_2847_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_2848_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_2849_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_2850_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_2851_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_2852_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_2853_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_2854_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_2855_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_2856_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_2857_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_2858_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_2859_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_2860_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_2861_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2862_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2863_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2864_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2865_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2866_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2867_sum__squares__eq__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),X3)),aa_real_real(times_times_real(Y),Y)) = zero_zero_real )
    <=> ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2868_sum__squares__eq__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2869_sum__squares__eq__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2870_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_2871_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2872_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2873_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_2874_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_2875_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2876_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_2877_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_2878_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_real ).

% power_zero_numeral
tff(fact_2879_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_int ).

% power_zero_numeral
tff(fact_2880_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2881_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_2882_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2883_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_2884_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_2885_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_2886_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_2887_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V2)) = $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,V2)),aa_num_real(numeral_numeral_real,V2),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_2888_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)) = $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,V2)),aa_num_rat(numeral_numeral_rat,V2),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_2889_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)) = $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,V2)),aa_num_nat(numeral_numeral_nat,V2),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_2890_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)) = $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,V2)),aa_num_int(numeral_numeral_int,V2),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_2891_max__0__1_I3_J,axiom,
    ! [X3: num] : aa_real_real(ord_max_real(zero_zero_real),aa_num_real(numeral_numeral_real,X3)) = aa_num_real(numeral_numeral_real,X3) ).

% max_0_1(3)
tff(fact_2892_max__0__1_I3_J,axiom,
    ! [X3: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,X3)) = aa_num_rat(numeral_numeral_rat,X3) ).

% max_0_1(3)
tff(fact_2893_max__0__1_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X3)) = aa_num_nat(numeral_numeral_nat,X3) ).

% max_0_1(3)
tff(fact_2894_max__0__1_I3_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X3)) = aa_num_int(numeral_numeral_int,X3) ).

% max_0_1(3)
tff(fact_2895_max__0__1_I4_J,axiom,
    ! [X3: num] : aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,X3)),zero_zero_real) = aa_num_real(numeral_numeral_real,X3) ).

% max_0_1(4)
tff(fact_2896_max__0__1_I4_J,axiom,
    ! [X3: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,X3)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X3) ).

% max_0_1(4)
tff(fact_2897_max__0__1_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X3)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X3) ).

% max_0_1(4)
tff(fact_2898_max__0__1_I4_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X3)),zero_zero_int) = aa_num_int(numeral_numeral_int,X3) ).

% max_0_1(4)
tff(fact_2899_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_2900_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2901_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2902_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_2903_max__0__1_I5_J,axiom,
    ! [X3: num] : aa_real_real(ord_max_real(one_one_real),aa_num_real(numeral_numeral_real,X3)) = aa_num_real(numeral_numeral_real,X3) ).

% max_0_1(5)
tff(fact_2904_max__0__1_I5_J,axiom,
    ! [X3: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,X3)) = aa_num_rat(numeral_numeral_rat,X3) ).

% max_0_1(5)
tff(fact_2905_max__0__1_I5_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X3)) = aa_num_nat(numeral_numeral_nat,X3) ).

% max_0_1(5)
tff(fact_2906_max__0__1_I5_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,X3)) = aa_num_int(numeral_numeral_int,X3) ).

% max_0_1(5)
tff(fact_2907_max__0__1_I6_J,axiom,
    ! [X3: num] : aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,X3)),one_one_real) = aa_num_real(numeral_numeral_real,X3) ).

% max_0_1(6)
tff(fact_2908_max__0__1_I6_J,axiom,
    ! [X3: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,X3)),one_one_rat) = aa_num_rat(numeral_numeral_rat,X3) ).

% max_0_1(6)
tff(fact_2909_max__0__1_I6_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X3)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X3) ).

% max_0_1(6)
tff(fact_2910_max__0__1_I6_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X3)),one_one_int) = aa_num_int(numeral_numeral_int,X3) ).

% max_0_1(6)
tff(fact_2911_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_2912_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_2913_mult__less__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_2914_nat__power__eq__Suc__0__iff,axiom,
    ! [X3: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( X3 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2915_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2916_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2917_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_2918_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_2919_nat__zero__less__power__iff,axiom,
    ! [X3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X3)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2920_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2921_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2922_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2923_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2924_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2925_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2926_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_2927_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_2928_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_2929_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_2930_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_2931_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_2932_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_2933_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_2934_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2935_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2936_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(power_power_real(A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2937_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2938_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2939_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_2940_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_2941_mult__le__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2942_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2943_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),Mb)),aa_nat_real(power_power_real(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2944_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2945_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2946_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2947_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_2948_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_2949_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_2950_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_2951_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_2952_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2953_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2954_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2955_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2956_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_2957_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_2958_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_2959_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_2960_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_2961_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2962_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_2963_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_2964_power2__eq__iff__nonneg,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2965_power2__eq__iff__nonneg,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2966_power2__eq__iff__nonneg,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2967_power2__eq__iff__nonneg,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2968_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),Mb)),aa_nat_real(power_power_real(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2969_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2970_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2971_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2972_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_2973_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_2974_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_2975_sum__power2__eq__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2976_sum__power2__eq__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2977_sum__power2__eq__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2978_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_2979_zero__reorient,axiom,
    ! [X3: literal] :
      ( ( zero_zero_literal = X3 )
    <=> ( X3 = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_2980_zero__reorient,axiom,
    ! [X3: real] :
      ( ( zero_zero_real = X3 )
    <=> ( X3 = zero_zero_real ) ) ).

% zero_reorient
tff(fact_2981_zero__reorient,axiom,
    ! [X3: rat] :
      ( ( zero_zero_rat = X3 )
    <=> ( X3 = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_2982_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
    <=> ( X3 = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_2983_zero__reorient,axiom,
    ! [X3: int] :
      ( ( zero_zero_int = X3 )
    <=> ( X3 = zero_zero_int ) ) ).

% zero_reorient
tff(fact_2984_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_2985_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ) ).

% max.mono
tff(fact_2986_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ) ).

% max.mono
tff(fact_2987_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_2988_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_2989_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.orderE
tff(fact_2990_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_2991_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_2992_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_2993_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_2994_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_2995_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_2996_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_2997_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.boundedE
tff(fact_2998_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.boundedE
tff(fact_2999_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.boundedE
tff(fact_3000_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.boundedE
tff(fact_3001_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3002_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3003_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3004_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3005_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.order_iff
tff(fact_3006_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_3007_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_3008_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_3009_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ).

% max.cobounded1
tff(fact_3010_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded1
tff(fact_3011_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_3012_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_3013_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ).

% max.cobounded2
tff(fact_3014_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded2
tff(fact_3015_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_3016_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_3017_le__max__iff__disj,axiom,
    ! [Z: rat,X3: rat,Y: 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,X3),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X3)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_3018_le__max__iff__disj,axiom,
    ! [Z: num,X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),X3)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_3019_le__max__iff__disj,axiom,
    ! [Z: nat,X3: nat,Y: 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,X3),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X3)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_3020_le__max__iff__disj,axiom,
    ! [Z: int,X3: int,Y: 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,X3),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X3)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_3021_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3022_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3023_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3024_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3025_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3026_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3027_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3028_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3029_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.coboundedI1
tff(fact_3030_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI1
tff(fact_3031_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_3032_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_3033_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.coboundedI2
tff(fact_3034_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI2
tff(fact_3035_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_3036_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_3037_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,A),B) = $ite(aa_set_nat_o(ord_less_eq_set_nat(A),B),B,A) ).

% max_def
tff(fact_3038_max__def,axiom,
    ! [A: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ).

% max_def
tff(fact_3039_max__def,axiom,
    ! [A: num,B: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ).

% max_def
tff(fact_3040_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_3041_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_3042_max__absorb1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_3043_max__absorb1,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_3044_max__absorb1,axiom,
    ! [Y: num,X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X3)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_3045_max__absorb1,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_3046_max__absorb1,axiom,
    ! [Y: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_3047_max__absorb2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_3048_max__absorb2,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_3049_max__absorb2,axiom,
    ! [X3: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_3050_max__absorb2,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_3051_max__absorb2,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_3052_max__add__distrib__left,axiom,
    ! [X3: real,Y: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(ord_max_real(X3),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(X3),Z)),aa_real_real(plus_plus_real(Y),Z)) ).

% max_add_distrib_left
tff(fact_3053_max__add__distrib__left,axiom,
    ! [X3: rat,Y: 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,X3),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_3054_max__add__distrib__left,axiom,
    ! [X3: nat,Y: 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,X3),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_3055_max__add__distrib__left,axiom,
    ! [X3: int,Y: 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,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_3056_max__add__distrib__right,axiom,
    ! [X3: real,Y: real,Z: real] : aa_real_real(plus_plus_real(X3),aa_real_real(ord_max_real(Y),Z)) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(X3),Y)),aa_real_real(plus_plus_real(X3),Z)) ).

% max_add_distrib_right
tff(fact_3057_max__add__distrib__right,axiom,
    ! [X3: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Z)) ).

% max_add_distrib_right
tff(fact_3058_max__add__distrib__right,axiom,
    ! [X3: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Z)) ).

% max_add_distrib_right
tff(fact_3059_max__add__distrib__right,axiom,
    ! [X3: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Z)) ).

% max_add_distrib_right
tff(fact_3060_max__diff__distrib__left,axiom,
    ! [X3: real,Y: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(ord_max_real(X3),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(X3),Z)),aa_real_real(minus_minus_real(Y),Z)) ).

% max_diff_distrib_left
tff(fact_3061_max__diff__distrib__left,axiom,
    ! [X3: rat,Y: 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,X3),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3062_max__diff__distrib__left,axiom,
    ! [X3: int,Y: 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,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3063_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_3064_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_3065_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(power_power_real(zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_3066_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_3067_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_3068_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_3069_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_3070_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(power_power_real(zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_3071_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_3072_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_3073_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q3)) ).

% nat_add_max_left
tff(fact_3074_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ).

% nat_add_max_right
tff(fact_3075_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).

% nat_mult_max_right
tff(fact_3076_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_max_left
tff(fact_3077_zero__le,axiom,
    ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X3) ).

% zero_le
tff(fact_3078_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_3079_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_3080_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_3081_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_3082_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_3083_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_3084_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_3085_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_3086_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_3087_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_3088_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_3089_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_3090_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_3091_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3092_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_3093_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_3094_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_3095_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_3096_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_3097_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_3098_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_3099_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_3100_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_3101_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_3102_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3103_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3104_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3105_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_3106_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_3107_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_3108_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_3109_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_3110_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_3111_vebt__buildup_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( X3 != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va3: nat] : X3 != aa_nat_nat(suc,aa_nat_nat(suc,Va3)) ) ) ).

% vebt_buildup.cases
tff(fact_3112_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N: nat] : X3 != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_3113_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_3114_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_3115_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_3116_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_3117_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_3118_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_3119_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X5: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X5),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X5),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X5)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_3120_zero__induct,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_3121_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_3122_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_3123_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_3124_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_3125_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_3126_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_3127_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_3128_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_3129_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_3130_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_3131_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_3132_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_3133_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_3134_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_3135_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_3136_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_3137_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_3138_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_3139_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_3140_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_3141_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(power_power_real(A),Nb) = aa_nat_real(power_power_real(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3142_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3143_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3144_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3145_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),Nb) = aa_nat_real(power_power_real(B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3146_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3147_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3148_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3149_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_3150_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3151_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3152_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3153_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3154_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_3155_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_3156_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_3157_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_3158_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_3159_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_3160_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_3161_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_3162_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3163_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3164_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3165_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3166_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_3167_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_3168_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_3169_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3170_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3171_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3172_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3173_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_3174_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3175_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3176_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_3177_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_3178_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3179_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3180_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_3181_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3182_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3183_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3184_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3185_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_3186_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_3187_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_3188_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_3189_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3190_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3191_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3192_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono
tff(fact_3193_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3194_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3195_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_3196_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3197_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3198_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3199_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono
tff(fact_3200_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3201_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3202_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_3203_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3204_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3205_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3206_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3207_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3208_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3209_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ).

% split_mult_pos_le
tff(fact_3210_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_3211_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_3212_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),A)) ).

% zero_le_square
tff(fact_3213_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_3214_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_3215_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3216_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3217_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3218_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3219_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3220_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3221_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3222_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3223_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_3224_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_3225_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_3226_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_3227_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_3228_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_3229_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_3230_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_3231_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing
tff(fact_3232_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3233_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3234_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3235_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_increasing
tff(fact_3236_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_3237_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_3238_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_3239_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3240_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3241_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3242_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3243_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_increasing2
tff(fact_3244_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_3245_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_3246_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_3247_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3248_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3249_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3250_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3251_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_3252_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_3253_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_3254_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_3255_add__nonneg__eq__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(plus_plus_real(X3),Y) = zero_zero_real )
        <=> ( ( X3 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3256_add__nonneg__eq__0__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y) = zero_zero_rat )
        <=> ( ( X3 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3257_add__nonneg__eq__0__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = zero_zero_nat )
        <=> ( ( X3 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3258_add__nonneg__eq__0__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y) = zero_zero_int )
        <=> ( ( X3 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3259_add__nonpos__eq__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(plus_plus_real(X3),Y) = zero_zero_real )
        <=> ( ( X3 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3260_add__nonpos__eq__0__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y) = zero_zero_rat )
        <=> ( ( X3 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3261_add__nonpos__eq__0__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = zero_zero_nat )
        <=> ( ( X3 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3262_add__nonpos__eq__0__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y) = zero_zero_int )
        <=> ( ( X3 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3263_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_3264_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_3265_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_3266_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_3267_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_3268_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_3269_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_3270_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_3271_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_3272_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_3273_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_3274_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_3275_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% pos_add_strict
tff(fact_3276_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_3277_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_3278_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_3279_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C5: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C5) )
           => ( C5 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3280_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_pos_pos
tff(fact_3281_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_3282_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_3283_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_3284_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_3285_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_3286_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_3287_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_3288_add__less__zeroD,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(X3),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_3289_add__less__zeroD,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_3290_add__less__zeroD,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_3291_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3292_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3293_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_3294_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3295_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3296_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_3297_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ).

% power_mono
tff(fact_3298_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_3299_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_3300_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_3301_zero__le__power,axiom,
    ! [A: real,Nb: 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(power_power_real(A),Nb)) ) ).

% zero_le_power
tff(fact_3302_zero__le__power,axiom,
    ! [A: rat,Nb: 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),Nb)) ) ).

% zero_le_power
tff(fact_3303_zero__le__power,axiom,
    ! [A: nat,Nb: 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),Nb)) ) ).

% zero_le_power
tff(fact_3304_zero__le__power,axiom,
    ! [A: int,Nb: 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),Nb)) ) ).

% zero_le_power
tff(fact_3305_zero__less__power,axiom,
    ! [A: real,Nb: 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(power_power_real(A),Nb)) ) ).

% zero_less_power
tff(fact_3306_zero__less__power,axiom,
    ! [A: rat,Nb: 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),Nb)) ) ).

% zero_less_power
tff(fact_3307_zero__less__power,axiom,
    ! [A: nat,Nb: 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),Nb)) ) ).

% zero_less_power
tff(fact_3308_zero__less__power,axiom,
    ! [A: int,Nb: 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),Nb)) ) ).

% zero_less_power
tff(fact_3309_length__pos__if__in__set,axiom,
    ! [X3: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3310_length__pos__if__in__set,axiom,
    ! [X3: real,Xs: list_real] :
      ( aa_set_real_o(member_real(X3),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_3311_length__pos__if__in__set,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X3,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_3312_length__pos__if__in__set,axiom,
    ! [X3: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((X3)),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_3313_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X3),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_3314_length__pos__if__in__set,axiom,
    ! [X3: int,Xs: list_int] :
      ( aa_set_int_o(member_int(X3),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_3315_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3316_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_3317_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3318_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3319_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3320_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_3321_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_3322_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
         => 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),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_3323_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M4: nat] : Nb = aa_nat_nat(suc,M4) ) ).

% gr0_conv_Suc
tff(fact_3324_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
          & 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),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_3325_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_3326_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_3327_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3328_option_Osize_I4_J,axiom,
    ! [X2: nat] : size_size_option_nat(some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3329_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3330_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
               => ~ aa_nat_o(P,I2) )
            & aa_nat_o(P,K) ) ) ) ).

% ex_least_nat_le
tff(fact_3331_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3332_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3333_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3334_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_3335_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_3336_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ) ).

% mult_less_mono2
tff(fact_3337_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ) ).

% mult_less_mono1
tff(fact_3338_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3339_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_3340_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_3341_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3342_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy,Uz)),X3) ).

% vebt_member.simps(3)
tff(fact_3343_vebt__insert_Osimps_I2_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,Sb: vEBT_VEBT,X3: nat] : vEBT_vebt_insert(vEBT_Node(Infoa,zero_zero_nat,Ts,Sb),X3) = vEBT_Node(Infoa,zero_zero_nat,Ts,Sb) ).

% vebt_insert.simps(2)
tff(fact_3344_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_3345_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3346_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3347_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3348_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3349_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3350_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3351_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3352_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3353_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3354_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3355_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3356_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3357_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3358_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3359_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3360_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3361_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3362_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3363_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3364_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3365_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3366_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3367_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3368_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3369_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3370_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3371_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3372_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3373_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3374_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3375_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3376_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3377_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3378_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3379_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3380_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3381_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3382_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3383_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3384_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3385_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3386_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3387_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3388_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3389_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3390_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3391_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3392_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3393_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3394_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3395_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_3396_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_3397_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_3398_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_3399_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3400_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3401_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3402_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3403_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_3404_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_3405_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_3406_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_3407_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3408_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3409_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3410_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3411_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing
tff(fact_3412_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3413_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3414_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3415_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3416_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3417_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3418_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3419_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3420_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3421_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_3422_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_3423_sum__squares__le__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),X3)),aa_real_real(times_times_real(Y),Y))),zero_zero_real)
    <=> ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3424_sum__squares__le__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3425_sum__squares__le__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3426_sum__squares__ge__zero,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),X3)),aa_real_real(times_times_real(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3427_sum__squares__ge__zero,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3428_sum__squares__ge__zero,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3429_mult__left__le__one__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Y),X3)),X3) ) ) ) ).

% mult_left_le_one_le
tff(fact_3430_mult__left__le__one__le,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X3)),X3) ) ) ) ).

% mult_left_le_one_le
tff(fact_3431_mult__left__le__one__le,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X3)),X3) ) ) ) ).

% mult_left_le_one_le
tff(fact_3432_mult__right__le__one__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(X3),Y)),X3) ) ) ) ).

% mult_right_le_one_le
tff(fact_3433_mult__right__le__one__le,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y)),X3) ) ) ) ).

% mult_right_le_one_le
tff(fact_3434_mult__right__le__one__le,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y)),X3) ) ) ) ).

% mult_right_le_one_le
tff(fact_3435_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3436_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3437_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3438_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3439_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),A) ) ) ).

% mult_left_le
tff(fact_3440_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3441_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3442_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3443_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3444_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3445_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3446_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3447_sum__squares__gt__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),X3)),aa_real_real(times_times_real(Y),Y)))
    <=> ( ( X3 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3448_sum__squares__gt__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( X3 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3449_sum__squares__gt__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( X3 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3450_not__sum__squares__lt__zero,axiom,
    ! [X3: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),X3)),aa_real_real(times_times_real(Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_3451_not__sum__squares__lt__zero,axiom,
    ! [X3: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_3452_not__sum__squares__lt__zero,axiom,
    ! [X3: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3453_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3454_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3455_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_3456_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_3457_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_3458_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_3459_power__le__one,axiom,
    ! [A: real,Nb: 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(power_power_real(A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_3460_power__le__one,axiom,
    ! [A: rat,Nb: 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),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3461_power__le__one,axiom,
    ! [A: nat,Nb: 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),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3462_power__le__one,axiom,
    ! [A: int,Nb: 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),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_3463_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3464_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3465_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3466_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3467_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3468_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3469_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X3) ).

% vebt_member.simps(4)
tff(fact_3470_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb)) = aa_nat_real(power_power_real(B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3471_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3472_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3473_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3474_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb))),aa_nat_real(power_power_real(B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3475_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3476_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3477_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3478_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3479_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3480_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3481_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3482_vebt__delete_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TrLst,Smry),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_3483_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_3484_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_3485_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3486_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3487_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3488_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_3489_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3490_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3491_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3492_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3493_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),Nb)
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K)
               => ~ aa_nat_o(P,I2) )
            & aa_nat_o(P,aa_nat_nat(suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_3494_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_3495_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => 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,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_3496_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_3497_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_3498_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_3499_power__gt__expt,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K2)) ) ).

% power_gt_expt
tff(fact_3500_nat__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_3501_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Mb)) ) ) ).

% div_le_mono2
tff(fact_3502_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_3503_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb)) ) ).

% nat_one_le_power
tff(fact_3504_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D3: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
           => aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split
tff(fact_3505_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D3: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
              & ~ aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split_asm
tff(fact_3506_vebt__insert_Osimps_I3_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,Sb: vEBT_VEBT,X3: nat] : vEBT_vebt_insert(vEBT_Node(Infoa,aa_nat_nat(suc,zero_zero_nat),Ts,Sb),X3) = vEBT_Node(Infoa,aa_nat_nat(suc,zero_zero_nat),Ts,Sb) ).

% vebt_insert.simps(3)
tff(fact_3507_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3508_vebt__delete_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X3: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),X3) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_3509_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3510_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3511_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3512_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3513_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3514_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3515_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3516_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3517_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3518_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3519_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3520_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3521_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3522_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3523_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3524_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3525_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3526_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3527_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3528_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3529_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3530_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3531_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3532_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3533_convex__bound__le,axiom,
    ! [X3: real,A: real,Y: real,U: real,V2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V2)
           => ( ( aa_real_real(plus_plus_real(U),V2) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),X3)),aa_real_real(times_times_real(V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3534_convex__bound__le,axiom,
    ! [X3: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V2)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = 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),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3535_convex__bound__le,axiom,
    ! [X3: int,A: int,Y: int,U: int,V2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V2)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = 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),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3536_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3537_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3538_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3539_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3540_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb))),aa_nat_real(power_power_real(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3541_power__Suc__less,axiom,
    ! [A: rat,Nb: 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),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3542_power__Suc__less,axiom,
    ! [A: nat,Nb: 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),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3543_power__Suc__less,axiom,
    ! [A: int,Nb: 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),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3544_power__Suc__le__self,axiom,
    ! [A: real,Nb: 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(power_power_real(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3545_power__Suc__le__self,axiom,
    ! [A: rat,Nb: 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,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3546_power__Suc__le__self,axiom,
    ! [A: nat,Nb: 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,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3547_power__Suc__le__self,axiom,
    ! [A: int,Nb: 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,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3548_power__Suc__less__one,axiom,
    ! [A: real,Nb: 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(power_power_real(A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_3549_power__Suc__less__one,axiom,
    ! [A: rat,Nb: 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,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3550_power__Suc__less__one,axiom,
    ! [A: nat,Nb: 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,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3551_power__Suc__less__one,axiom,
    ! [A: int,Nb: 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,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3552_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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(power_power_real(A),N4)),aa_nat_real(power_power_real(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3553_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3554_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3555_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3556_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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(power_power_real(A),N4)),aa_nat_real(power_power_real(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3557_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,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),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3558_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3559_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3560_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_3561_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_3562_zero__power2,axiom,
    aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_3563_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_3564_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_3565_self__le__power,axiom,
    ! [A: real,Nb: 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),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(power_power_real(A),Nb)) ) ) ).

% self_le_power
tff(fact_3566_self__le__power,axiom,
    ! [A: rat,Nb: 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),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_3567_self__le__power,axiom,
    ! [A: nat,Nb: 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),Nb)
       => 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),Nb)) ) ) ).

% self_le_power
tff(fact_3568_self__le__power,axiom,
    ! [A: int,Nb: 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),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% self_le_power
tff(fact_3569_one__less__power,axiom,
    ! [A: real,Nb: 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),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),Nb)) ) ) ).

% one_less_power
tff(fact_3570_one__less__power,axiom,
    ! [A: rat,Nb: 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),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_3571_one__less__power,axiom,
    ! [A: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_3572_one__less__power,axiom,
    ! [A: int,Nb: 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),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% one_less_power
tff(fact_3573_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_3574_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_3575_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3576_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3577_power__diff,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3578_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3579_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3580_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_3581_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3582_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3583_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3584_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3585_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_3586_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I4)),J3) )
               => aa_nat_o(P,I4) ) ) ) ) ) ).

% split_div
tff(fact_3587_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_3588_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_3589_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_3590_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,Va,Vb)),X3)
    <=> ( ( X3 = Mia )
        | ( X3 = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3591_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_3592_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc,Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3593_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3594_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3595_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3596_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3597_convex__bound__lt,axiom,
    ! [X3: real,A: real,Y: real,U: real,V2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V2)
           => ( ( aa_real_real(plus_plus_real(U),V2) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),X3)),aa_real_real(times_times_real(V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3598_convex__bound__lt,axiom,
    ! [X3: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V2)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = 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),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3599_convex__bound__lt,axiom,
    ! [X3: int,A: int,Y: int,U: int,V2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V2)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = 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),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3600_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3601_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_3602_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_3603_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_3604_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3605_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_3606_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_3607_power2__eq__imp__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3608_power2__eq__imp__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3609_power2__eq__imp__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3610_power2__eq__imp__eq,axiom,
    ! [X3: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3611_power2__le__imp__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ) ).

% power2_le_imp_le
tff(fact_3612_power2__le__imp__le,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ) ).

% power2_le_imp_le
tff(fact_3613_power2__le__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ).

% power2_le_imp_le
tff(fact_3614_power2__le__imp__le,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ) ).

% power2_le_imp_le
tff(fact_3615_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_3616_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_3617_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_3618_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3619_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3620_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3621_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3622_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3623_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3624_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3625_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),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,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3626_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_3627_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_3628_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_3629_power__eq__if,axiom,
    ! [P2: complex,Mb: nat] :
      aa_nat_complex(power_power_complex(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3630_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(power_power_real(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(times_times_real(P2),aa_nat_real(power_power_real(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3631_power__eq__if,axiom,
    ! [P2: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3632_power__eq__if,axiom,
    ! [P2: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3633_power__eq__if,axiom,
    ! [P2: int,Mb: nat] :
      aa_nat_int(power_power_int(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3634_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3635_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(power_power_real(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3636_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3637_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3638_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3639_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_3640_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_3641_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_3642_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3643_power2__less__imp__less,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ) ).

% power2_less_imp_less
tff(fact_3644_power2__less__imp__less,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y) ) ) ).

% power2_less_imp_less
tff(fact_3645_power2__less__imp__less,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ) ).

% power2_less_imp_less
tff(fact_3646_power2__less__imp__less,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y) ) ) ).

% power2_less_imp_less
tff(fact_3647_sum__power2__ge__zero,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3648_sum__power2__ge__zero,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3649_sum__power2__ge__zero,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3650_sum__power2__le__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3651_sum__power2__le__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3652_sum__power2__le__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3653_not__sum__power2__lt__zero,axiom,
    ! [X3: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_3654_not__sum__power2__lt__zero,axiom,
    ! [X3: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_3655_not__sum__power2__lt__zero,axiom,
    ! [X3: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_3656_sum__power2__gt__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X3 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3657_sum__power2__gt__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X3 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3658_sum__power2__gt__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X3 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3659_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3660_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: 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))),Nb))) ).

% zero_le_even_power'
tff(fact_3661_zero__le__even__power_H,axiom,
    ! [A: int,Nb: 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))),Nb))) ).

% zero_le_even_power'
tff(fact_3662_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_3663_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_3664_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_3665_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B4)
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),set_complex2(Xs))
         => aa_set_complex_o(member_complex(X4),B4) ) ) ).

% subset_code(1)
tff(fact_3666_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B4)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),set_real2(Xs))
         => aa_set_real_o(member_real(X4),B4) ) ) ).

% subset_code(1)
tff(fact_3667_subset__code_I1_J,axiom,
    ! [Xs: list_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(set_o2(Xs)),B4)
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),set_o2(Xs))
         => aa_set_o_o(member_o((X4)),B4) ) ) ).

% subset_code(1)
tff(fact_3668_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_code(1)
tff(fact_3669_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B4)
    <=> ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X4,B4) ) ) ).

% subset_code(1)
tff(fact_3670_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_code(1)
tff(fact_3671_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_3672_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_3673_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_3674_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_3675_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => 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_3676_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: 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))),Nb))))
     => 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_3677_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: 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))),Nb))))
     => 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_3678_odd__power__less__zero,axiom,
    ! [A: real,Nb: 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(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))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_3679_odd__power__less__zero,axiom,
    ! [A: rat,Nb: 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))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_3680_odd__power__less__zero,axiom,
    ! [A: int,Nb: 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))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_3681_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X3: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3682_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X3: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_3683_arith__geo__mean,axiom,
    ! [U: real,X3: real,Y: real] :
      ( ( aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(X3),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3684_arith__geo__mean,axiom,
    ! [U: rat,X3: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3685_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3686_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3687_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3688_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3689_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K2 )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),nth_VEBT_VEBT(Xs3,I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3690_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_o_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I4),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K2 )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
             => aa_o_o(aa_nat_fun_o_o(P,I4),aa_nat_o(nth_o(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3691_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I4),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K2 )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
             => aa_nat_o(aa_nat_fun_nat_o(P,I4),aa_nat_nat(nth_nat(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3692_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_int_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I4),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K2 )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
             => aa_int_o(aa_nat_fun_int_o(P,I4),aa_nat_int(nth_int(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3693_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [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(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3694_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [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(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3695_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [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(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3696_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [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(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3697_set__update__subsetI,axiom,
    ! [Xs: list_complex,A4: set_complex,X3: complex,I: nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,X3))),A4) ) ) ).

% set_update_subsetI
tff(fact_3698_set__update__subsetI,axiom,
    ! [Xs: list_real,A4: set_real,X3: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X3))),A4) ) ) ).

% set_update_subsetI
tff(fact_3699_set__update__subsetI,axiom,
    ! [Xs: list_o,A4: set_o,X3: $o,I: nat] :
      ( aa_set_o_o(ord_less_eq_set_o(set_o2(Xs)),A4)
     => ( aa_set_o_o(member_o((X3)),A4)
       => aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,I,(X3)))),A4) ) ) ).

% set_update_subsetI
tff(fact_3700_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,X3: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A4)
     => ( aa_set_int_o(member_int(X3),A4)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X3))),A4) ) ) ).

% set_update_subsetI
tff(fact_3701_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A4)
     => ( member_VEBT_VEBT(X3,A4)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X3)),A4) ) ) ).

% set_update_subsetI
tff(fact_3702_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,X3: nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A4)
     => ( aa_set_nat_o(member_nat(X3),A4)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X3))),A4) ) ) ).

% set_update_subsetI
tff(fact_3703_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3704_vebt__delete_Osimps_I4_J,axiom,
    ! [Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu2: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya),Uu2) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ).

% vebt_delete.simps(4)
tff(fact_3705_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),X3) ).

% vebt_member.simps(2)
tff(fact_3706_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_3707_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ! [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_3708_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),set_o2(Xs))
         => aa_o_o(P,(X4)) )
    <=> ! [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_3709_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
    <=> ! [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_3710_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_int_o(P,X4) )
    <=> ! [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_3711_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,X3: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I3)) )
     => ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
       => aa_complex_o(P,X3) ) ) ).

% all_nth_imp_all_set
tff(fact_3712_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,X3: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I3)) )
     => ( aa_set_real_o(member_real(X3),set_real2(Xs))
       => aa_real_o(P,X3) ) ) ).

% all_nth_imp_all_set
tff(fact_3713_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,X3: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) )
     => ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,X3) ) ) ).

% all_nth_imp_all_set
tff(fact_3714_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,X3: $o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
     => ( aa_set_o_o(member_o((X3)),set_o2(Xs))
       => aa_o_o(P,(X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_3715_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,X3: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
     => ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
       => aa_nat_o(P,X3) ) ) ).

% all_nth_imp_all_set
tff(fact_3716_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,X3: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
     => ( aa_set_int_o(member_int(X3),set_int2(Xs))
       => aa_int_o(P,X3) ) ) ).

% all_nth_imp_all_set
tff(fact_3717_in__set__conv__nth,axiom,
    ! [X3: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I4) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_3718_in__set__conv__nth,axiom,
    ! [X3: real,Xs: list_real] :
      ( aa_set_real_o(member_real(X3),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) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_3719_in__set__conv__nth,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X3,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) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_3720_in__set__conv__nth,axiom,
    ! [X3: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((X3)),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)
          <=> (X3) ) ) ) ).

% in_set_conv_nth
tff(fact_3721_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X3),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) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_3722_in__set__conv__nth,axiom,
    ! [X3: int,Xs: list_int] :
      ( aa_set_int_o(member_int(X3),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) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_3723_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X5: vEBT_VEBT] :
            ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X5) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_3724_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),set_o2(Xs))
           => aa_o_o(P,(X5)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3725_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
           => aa_nat_o(P,X5) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3726_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),set_int2(Xs))
           => aa_int_o(P,X5) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3727_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_3728_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_3729_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_3730_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o(aa_nat_o(nth_o(Xs),Nb)),set_o2(Xs)) ) ).

% nth_mem
tff(fact_3731_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3732_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_3733_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X3: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X3))),insert_int(X3,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_3734_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,X3: $o] : aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,I,(X3)))),insert_o((X3),set_o2(Xs))) ).

% set_update_subset_insert
tff(fact_3735_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X3: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X3))),insert8211810215607154385at_nat(X3,set_Pr5648618587558075414at_nat(Xs))) ).

% set_update_subset_insert
tff(fact_3736_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I: nat,X3: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,I,X3))),insert9069300056098147895at_nat(X3,set_Pr3765526544606949372at_nat(Xs))) ).

% set_update_subset_insert
tff(fact_3737_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X3)),insert_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_3738_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X3))),insert_nat(X3,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_3739_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,X3: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(X3),set_complex2(list_update_complex(Xs,Nb,X3))) ) ).

% set_update_memI
tff(fact_3740_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(X3),set_real2(list_update_real(Xs,Nb,X3))) ) ).

% set_update_memI
tff(fact_3741_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(X3,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,X3))) ) ).

% set_update_memI
tff(fact_3742_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,X3: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o((X3)),set_o2(list_update_o(Xs,Nb,(X3)))) ) ).

% set_update_memI
tff(fact_3743_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(X3),set_nat2(list_update_nat(Xs,Nb,X3))) ) ).

% set_update_memI
tff(fact_3744_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,X3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(X3),set_int2(list_update_int(Xs,Nb,X3))) ) ).

% set_update_memI
tff(fact_3745_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: 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,X3),J) = $ite(I = J,X3,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_3746_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X3: $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,(X3))),J)
      <=> $ite(I = J,(X3),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3747_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,X3: 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,X3)),J) = $ite(I = J,X3,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3748_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,X3: 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,X3)),J) = $ite(I = J,X3,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3749_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X3) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_3750_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X3: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(X3)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (X3) ) ) ) ).

% list_update_same_conv
tff(fact_3751_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,X3) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_3752_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,X3) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_3753_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X3: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),X3) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia),Mia,X3),
        $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)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( X3 = Mia )
                | ( X3 = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Mia),X3,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3754_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_3755_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_3756_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_3757_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_3758_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_3759_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_3760_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_3761_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_3762_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_3763_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_3764_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_3765_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_3766_vebt__pred_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X3,Xa2) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : X3 = vEBT_Leaf((Uu),(Uv))
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw: $o] : X3 = vEBT_Leaf((A3),(Uw))
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3 = vEBT_Leaf((A3),(B3)) )
               => ( ? [Va3: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3))
                 => ( Y != $ite(
                        (B3),
                        some_nat(one_one_nat),
                        $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2)
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                              some_nat(Ma),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa2),some_nat(Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_3767_vebt__succ_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X3,Xa2) = Y )
     => ( ! [Uu: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((Uu),(B3)) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y != $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : X3 = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xa2 = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                            some_nat(Mi),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_3768_vebt__delete_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X3,Xa2) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B3)) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] : X3 = vEBT_Leaf((A3),(B3))
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A3),$false) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3 = vEBT_Leaf((A3),(B3)) )
               => ( ? [N: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y != vEBT_Leaf((A3),(B3)) ) ) )
           => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) ) )
             => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2) ) )
               => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),
                              $ite(
                                ( ( Xa2 = Mi )
                                & ( Xa2 = Ma ) ),
                                vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xa2 = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xa2),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xa2 = Mi,xn,Mi),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                      $ite(
                                                        xn = Ma,
                                                        $let(
                                                          maxs: option_nat,
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                        Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),newlist,sn) ),
                                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),newlist,Summary2) ) ) ),
                                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_3769_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_3770_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_3771_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ).

% deg_1_Leaf
tff(fact_3772_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A6: $o,B7: $o] : Ta = vEBT_Leaf((A6),(B7)) ) ).

% deg1Leaf
tff(fact_3773_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_3774_half__nonnegative__int__iff,axiom,
    ! [K2: 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,K2),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),K2) ) ).

% half_nonnegative_int_iff
tff(fact_3775_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),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,K2),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_3776_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_3777_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_3778_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_3779_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X5)
     => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(Info,Deg,TreeList,Summary2),X5) ) ).

% VEBT_internal.insert'.cases
tff(fact_3780_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_3781_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv2))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_3782_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu2: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu2),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_3783_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu2),(Uv2))),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_3784_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = vEBT_Leaf((A),(B)) ).

% vebt_delete.simps(3)
tff(fact_3785_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_3786_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu2),(Uv2))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_3787_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] : vEBT_VEBT_insert(vEBT_Leaf((A),(B)),X3) = vEBT_vebt_insert(vEBT_Leaf((A),(B)),X3) ).

% VEBT_internal.insert'.simps(1)
tff(fact_3788_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X5)
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S),X5) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3789_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_3790_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A),$false) ).

% vebt_delete.simps(2)
tff(fact_3791_VEBT__internal_OminNull_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( X3 != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : X3 != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : X3 != vEBT_Leaf((Uu),$true)
         => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3792_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),X3)
    <=> $ite(
          X3 = zero_zero_nat,
          (A),
          $ite(X3 = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_3793_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_3794_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X3)
     => ( ! [Uv: $o] : X3 != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : X3 != vEBT_Leaf((Uu),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3795_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),X3)
    <=> $ite(
          X3 = zero_zero_nat,
          (A),
          $ite(X3 = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3796_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3797_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_3798_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X3)
     => ( ( X3 != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3799_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3800_realpow__pos__nth2,axiom,
    ! [A: real,Nb: 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(power_power_real(R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3801_real__arch__pow__inv,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X3),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_3802_vebt__mint_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : X3 != vEBT_Leaf((A3),(B3))
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_3803_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X3)
      <=> (Y) )
     => ( ( ( X3 = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv: $o] : X3 = vEBT_Leaf($true,(Uv))
           => (Y) )
         => ( ( ? [Uu: $o] : X3 = vEBT_Leaf((Uu),$true)
             => (Y) )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3804_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(power_power_real(R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3805_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X5)
            & ( aa_nat_real(power_power_real(X5),Nb) = A )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y5)
                  & ( aa_nat_real(power_power_real(Y5),Nb) = A ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3806_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3807_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3808_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
        (A),
        some_nat(zero_zero_nat),
        $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_3809_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_3810_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A),(Uw2)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_3811_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu2),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_3812_int__power__div__base,axiom,
    ! [Mb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K2),Mb)),K2) = aa_nat_int(power_power_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3813_vebt__pred_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)
     => ( ! [A3: $o,Uw: $o] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B3: $o,Va3: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2),Vb2)
           => ( ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),X5) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_3814_vebt__succ_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B3: $o] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)
     => ( ! [Uv: $o,Uw: $o,N: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va2)
         => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),X5) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_3815_vebt__insert_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X5)
     => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts2,S),X5)
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X5)
         => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2),X5)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),X5) ) ) ) ) ).

% vebt_insert.cases
tff(fact_3816_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Uw)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),X5)
         => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2),X5)
           => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2),X5) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3817_vebt__delete_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),zero_zero_nat)
     => ( ! [A3: $o,B3: $o] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B3: $o,N: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2),Uu)
           => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2),X5)
             => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),X5)
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),X5) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_3818_vebt__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X5)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),X5)
       => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2),X5)
         => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X5)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X3 != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),X5) ) ) ) ) ).

% vebt_member.cases
tff(fact_3819_vebt__pred_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_3820_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_3821_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% add_divide_distrib
tff(fact_3822_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ).

% add_divide_distrib
tff(fact_3823_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X3,Xa2) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf((A3),(B3)),Xa2) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(Info,Deg,TreeList,Summary2) )
             => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),Xa2),vEBT_Node(Info,Deg,TreeList,Summary2),vEBT_vebt_insert(vEBT_Node(Info,Deg,TreeList,Summary2),Xa2)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_3824_vebt__mint_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X3) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (A3),
                  some_nat(zero_zero_nat),
                  $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3825_vebt__maxt_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X3) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (B3),
                  some_nat(one_one_nat),
                  $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3826_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3827_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3828_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_right_mono
tff(fact_3829_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) ) ) ).

% divide_right_mono
tff(fact_3830_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_3831_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3832_divide__nonneg__nonneg,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X3),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3833_divide__nonneg__nonneg,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3834_divide__nonneg__nonpos,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_3835_divide__nonneg__nonpos,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3836_divide__nonpos__nonneg,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_3837_divide__nonpos__nonneg,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3838_divide__nonpos__nonpos,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X3),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3839_divide__nonpos__nonpos,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3840_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(divide_divide_real(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3841_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3842_field__le__epsilon,axiom,
    ! [X3: real,Y: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(plus_plus_real(Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ).

% field_le_epsilon
tff(fact_3843_field__le__epsilon,axiom,
    ! [X3: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ).

% field_le_epsilon
tff(fact_3844_divide__nonpos__pos,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3845_divide__nonpos__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3846_divide__nonpos__neg,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X3),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3847_divide__nonpos__neg,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3848_divide__nonneg__pos,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X3),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3849_divide__nonneg__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3850_divide__nonneg__neg,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3851_divide__nonneg__neg,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3852_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3853_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3854_frac__less2,axiom,
    ! [X3: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X3),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3855_frac__less2,axiom,
    ! [X3: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3856_frac__less,axiom,
    ! [X3: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X3),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3857_frac__less,axiom,
    ! [X3: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3858_frac__le,axiom,
    ! [Y: real,X3: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3859_frac__le,axiom,
    ! [Y: rat,X3: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3860_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3861_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3862_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3863_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),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)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3864_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3865_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3866_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X3: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X3),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X3),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3867_add__frac__eq,axiom,
    ! [Y: real,Z: real,X3: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X3),Y)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3868_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3869_add__frac__num,axiom,
    ! [Y: complex,X3: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X3),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X3),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3870_add__frac__num,axiom,
    ! [Y: real,X3: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X3),Y)),Z) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3871_add__frac__num,axiom,
    ! [Y: rat,X3: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3872_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X3: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X3),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X3),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3873_add__num__frac,axiom,
    ! [Y: real,Z: real,X3: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(X3),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3874_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X3: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3875_add__divide__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X3),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X3),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3876_add__divide__eq__iff,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(X3),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3877_add__divide__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3878_divide__add__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X3),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X3),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3879_divide__add__eq__iff,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X3),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3880_divide__add__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3881_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_3882_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_3883_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_3884_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_3885_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X3),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xa2 = zero_zero_nat,
                (A3),
                $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 != vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
         => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : X3 = vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3886_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X3),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xa2 = zero_zero_nat,
                  (A3),
                  $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : X3 = vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3887_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X3),Xa2)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa2 = zero_zero_nat,
                    (A3),
                    $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 = vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : X3 = vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3888_vebt__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_vebt_member(X3),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xa2 = zero_zero_nat,
                  (A3),
                  $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2)
             => ~ $ite(
                    Xa2 = Mi,
                    $true,
                    $ite(
                      Xa2 = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3889_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X3),Xa2)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : X3 = vEBT_Leaf((Uu),(Uv))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2)
               => ( (Y)
                <=> ~ ( ( Xa2 = Mi )
                      | ( Xa2 = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xa2 = Mi )
                        | ( Xa2 = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3890_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X3),Xa2)
     => ( ! [Uu: $o,Uv: $o] : X3 != vEBT_Leaf((Uu),(Uv))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2)
               => ( ( Xa2 = Mi )
                  | ( Xa2 = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2)
                 => ( ( Xa2 = Mi )
                    | ( Xa2 = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3891_field__le__mult__one__interval,axiom,
    ! [X3: real,Y: 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(times_times_real(Z2),X3)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ).

% field_le_mult_one_interval
tff(fact_3892_field__le__mult__one__interval,axiom,
    ! [X3: rat,Y: 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),X3)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ).

% field_le_mult_one_interval
tff(fact_3893_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3894_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3895_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Y)),X3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(divide_divide_real(X3),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3896_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3897_mult__imp__div__pos__le,axiom,
    ! [Y: real,X3: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3898_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X3: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3899_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3900_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3901_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3902_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3903_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3904_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3905_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3906_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3907_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3908_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3909_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3910_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3911_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_3912_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3913_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3914_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3915_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3916_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3917_frac__le__eq,axiom,
    ! [Y: real,Z: real,X3: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X3),Y)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X3),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_3918_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3919_vebt__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(X3),Xa2)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa2 = zero_zero_nat,
                    (A3),
                    $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => (Y) )
         => ( ( ? [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xa2 = Mi,
                            $true,
                            $ite(
                              Xa2 = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3920_vebt__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(X3),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xa2 = zero_zero_nat,
                (A3),
                $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X3 != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
         => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 != vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2)
           => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 != vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2)
                   => $ite(
                        Xa2 = Mi,
                        $true,
                        $ite(
                          Xa2 = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3921_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A3: $o,B3: $o] : A1 = vEBT_Leaf((A3),(B3))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                       => ( ( M = aa_nat_nat(suc,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                         => ( ( M = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                             => ( ! [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))),M))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I2) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [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))),M))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I2 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary2) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary2),M)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                           => ( ( M = aa_nat_nat(suc,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                               => ( ! [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))),M))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I2) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [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))),M))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I2 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3922_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A6: $o,B7: $o] : A1 = vEBT_Leaf((A6),(B7))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N3)
            & ( 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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N3))
            & ( 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,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N3)
            & ( 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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [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))),N3))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [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))),N3))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N3) = I4 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N3)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma2) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N3))
            & ( 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,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [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,N3)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [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,N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N3) = I4 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N3)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3923_vebt__insert_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X3,Xa2) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  Xa2 = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xa2 = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(Info,zero_zero_nat,Ts2,S) )
             => ( Y != vEBT_Node(Info,zero_zero_nat,Ts2,S) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
               => ( Y != vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ) )
           => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),Mi,Xa2),
                          $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)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xa2 = Mi )
                                  | ( Xa2 = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),Xa2,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(Summary2,h),Summary2)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3924_scaling__mono,axiom,
    ! [U: real,V2: real,R2: real,Sb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V2)
     => ( 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),Sb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),aa_real_real(minus_minus_real(V2),U))),Sb))),V2) ) ) ) ).

% scaling_mono
tff(fact_3925_scaling__mono,axiom,
    ! [U: rat,V2: rat,R2: rat,Sb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V2)
     => ( 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),Sb)
         => 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,V2),U))),Sb))),V2) ) ) ) ).

% scaling_mono
tff(fact_3926_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_3927_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_3928_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(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_3929_vebt__succ_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X3,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(B3)) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = $ite((B3),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X3 = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xa2 = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa2)) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd2),Xa2)) ) )
               => ( ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa2)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                                some_nat(Mi),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3930_vebt__pred_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X3,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X3 = vEBT_Leaf((A3),(Uw)) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A3),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X3 = vEBT_Leaf((A3),(B3)) )
                 => ! [Va3: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                     => ( ( Y = $ite(
                              (B3),
                              some_nat(one_one_nat),
                              $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2),Xa2)) ) )
               => ( ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2,Ve2),Xa2)) ) )
                 => ( ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa2)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                                  some_nat(Ma),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa2),some_nat(Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_3931_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_3932_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_3933_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_3934_vebt__delete_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X3,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B3)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3 = vEBT_Leaf((A3),(B3)) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A3),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X3 = vEBT_Leaf((A3),(B3)) )
                 => ! [N: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf((A3),(B3)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary2),Xa2)) ) )
               => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2),Xa2)) ) )
                 => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xa2)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),
                                  $ite(
                                    ( ( Xa2 = Mi )
                                    & ( Xa2 = Ma ) ),
                                    vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xa2 = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xa2),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xa2 = Mi,xn,Mi),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary2,h),
                                                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                          $ite(
                                                            xn = Ma,
                                                            $let(
                                                              maxs: option_nat,
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                            Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),newlist,sn) ),
                                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),newlist,Summary2) ) ) ),
                                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_3935_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_3936_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(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_3937_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
          & aa_nat_o(P,M4) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less
tff(fact_3938_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
         => aa_nat_o(P,M4) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less
tff(fact_3939_vebt__insert_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X3,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      Xa2 = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xa2 = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Info,zero_zero_nat,Ts2,S) )
               => ( ( Y = vEBT_Node(Info,zero_zero_nat,Ts2,S) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts2,S),Xa2)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                 => ( ( Y = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xa2)) ) )
             => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2),Xa2)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),Mi,Xa2),
                              $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)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xa2 = Mi )
                                      | ( Xa2 = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),Xa2,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(Summary2,h),Summary2)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3940_insert_H__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,X3: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(vEBT_VEBT_insert(Ta,X3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(Ta)),insert_nat(X3,bot_bot_set_nat))),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ) ).

% insert'_correct
tff(fact_3941_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X3,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf((A3),(B3)),Xa2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2)) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Info,Deg,TreeList,Summary2) )
               => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),Xa2),vEBT_Node(Info,Deg,TreeList,Summary2),vEBT_vebt_insert(vEBT_Node(Info,Deg,TreeList,Summary2),Xa2)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,Deg,TreeList,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_3942_vebt__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2))
               => $ite(
                    Xa2 = zero_zero_nat,
                    (A3),
                    $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa2)) )
           => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2))
                       => $ite(
                            Xa2 = Mi,
                            $true,
                            $ite(
                              Xa2 = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3943_vebt__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(X3),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xa2 = zero_zero_nat,
                      (A3),
                      $ite(Xa2 = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa2)) ) )
           => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xa2 = Mi,
                              $true,
                              $ite(
                                Xa2 = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3944_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X3),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xa2 = zero_zero_nat,
                      (A3),
                      $ite(Xa2 = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3945_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2))
               => ~ $ite(
                      Xa2 = zero_zero_nat,
                      (A3),
                      $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S),Xa2))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3946_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2))
               => $ite(
                    Xa2 = zero_zero_nat,
                    (A3),
                    $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList,S),Xa2))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3947_IntI,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),A4)
     => ( aa_set_complex_o(member_complex(C),B4)
       => aa_set_complex_o(member_complex(C),inf_inf_set_complex(A4,B4)) ) ) ).

% IntI
tff(fact_3948_IntI,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),A4)
     => ( aa_set_real_o(member_real(C),B4)
       => aa_set_real_o(member_real(C),inf_inf_set_real(A4,B4)) ) ) ).

% IntI
tff(fact_3949_IntI,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),A4)
     => ( aa_set_o_o(member_o((C)),B4)
       => aa_set_o_o(member_o((C)),inf_inf_set_o(A4,B4)) ) ) ).

% IntI
tff(fact_3950_IntI,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A4)
     => ( aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),inf_inf_set_int(A4,B4)) ) ) ).

% IntI
tff(fact_3951_IntI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A4)
     => ( aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ) ) ).

% IntI
tff(fact_3952_IntI,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A4)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B4)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ) ) ).

% IntI
tff(fact_3953_Int__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A4,B4))
    <=> ( aa_set_complex_o(member_complex(C),A4)
        & aa_set_complex_o(member_complex(C),B4) ) ) ).

% Int_iff
tff(fact_3954_Int__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A4,B4))
    <=> ( aa_set_real_o(member_real(C),A4)
        & aa_set_real_o(member_real(C),B4) ) ) ).

% Int_iff
tff(fact_3955_Int__iff,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A4,B4))
    <=> ( aa_set_o_o(member_o((C)),A4)
        & aa_set_o_o(member_o((C)),B4) ) ) ).

% Int_iff
tff(fact_3956_Int__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A4,B4))
    <=> ( aa_set_int_o(member_int(C),A4)
        & aa_set_int_o(member_int(C),B4) ) ) ).

% Int_iff
tff(fact_3957_Int__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
    <=> ( aa_set_nat_o(member_nat(C),A4)
        & aa_set_nat_o(member_nat(C),B4) ) ) ).

% Int_iff
tff(fact_3958_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A4)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B4) ) ) ).

% Int_iff
tff(fact_3959_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_3960_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% inf.bounded_iff
tff(fact_3961_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_3962_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_3963_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.bounded_iff
tff(fact_3964_le__inf__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Z) ) ) ).

% le_inf_iff
tff(fact_3965_le__inf__iff,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
        & aa_set_nat_o(ord_less_eq_set_nat(X3),Z) ) ) ).

% le_inf_iff
tff(fact_3966_le__inf__iff,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) ) ) ).

% le_inf_iff
tff(fact_3967_le__inf__iff,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z) ) ) ).

% le_inf_iff
tff(fact_3968_le__inf__iff,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z) ) ) ).

% le_inf_iff
tff(fact_3969_inf__bot__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% inf_bot_right
tff(fact_3970_inf__bot__right,axiom,
    ! [X3: set_o] : inf_inf_set_o(X3,bot_bot_set_o) = bot_bot_set_o ).

% inf_bot_right
tff(fact_3971_inf__bot__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_3972_inf__bot__right,axiom,
    ! [X3: set_int] : inf_inf_set_int(X3,bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_3973_inf__bot__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X3) = bot_bo2099793752762293965at_nat ).

% inf_bot_left
tff(fact_3974_inf__bot__left,axiom,
    ! [X3: set_o] : inf_inf_set_o(bot_bot_set_o,X3) = bot_bot_set_o ).

% inf_bot_left
tff(fact_3975_inf__bot__left,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X3) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_3976_inf__bot__left,axiom,
    ! [X3: set_int] : inf_inf_set_int(bot_bot_set_int,X3) = bot_bot_set_int ).

% inf_bot_left
tff(fact_3977_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A4)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B4) ) ) ).

% Int_subset_iff
tff(fact_3978_Int__subset__iff,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
        & aa_set_nat_o(ord_less_eq_set_nat(C2),B4) ) ) ).

% Int_subset_iff
tff(fact_3979_Int__insert__right__if1,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(A,B4)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3980_Int__insert__right__if1,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(A),A4)
     => ( inf_inf_set_complex(A4,insert_complex(A,B4)) = insert_complex(A,inf_inf_set_complex(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3981_Int__insert__right__if1,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ( inf_inf_set_real(A4,insert_real(A,B4)) = insert_real(A,inf_inf_set_real(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3982_Int__insert__right__if1,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( inf_inf_set_o(A4,insert_o((A),B4)) = insert_o((A),inf_inf_set_o(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3983_Int__insert__right__if1,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( inf_inf_set_int(A4,insert_int(A,B4)) = insert_int(A,inf_inf_set_int(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3984_Int__insert__right__if1,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(A,B4)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3985_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(A,B4)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3986_Int__insert__right__if0,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(A,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_3987_Int__insert__right__if0,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A),A4)
     => ( inf_inf_set_complex(A4,insert_complex(A,B4)) = inf_inf_set_complex(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_3988_Int__insert__right__if0,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(A),A4)
     => ( inf_inf_set_real(A4,insert_real(A,B4)) = inf_inf_set_real(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_3989_Int__insert__right__if0,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A4)
     => ( inf_inf_set_o(A4,insert_o((A),B4)) = inf_inf_set_o(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_3990_Int__insert__right__if0,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),A4)
     => ( inf_inf_set_int(A4,insert_int(A,B4)) = inf_inf_set_int(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_3991_Int__insert__right__if0,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_3992_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(A,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_3993_insert__inter__insert,axiom,
    ! [A: int,A4: set_int,B4: set_int] : inf_inf_set_int(insert_int(A,A4),insert_int(A,B4)) = insert_int(A,inf_inf_set_int(A4,B4)) ).

% insert_inter_insert
tff(fact_3994_insert__inter__insert,axiom,
    ! [A: $o,A4: set_o,B4: set_o] : inf_inf_set_o(insert_o((A),A4),insert_o((A),B4)) = insert_o((A),inf_inf_set_o(A4,B4)) ).

% insert_inter_insert
tff(fact_3995_insert__inter__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A4)),insert9069300056098147895at_nat(A,B4)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) ).

% insert_inter_insert
tff(fact_3996_insert__inter__insert,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A4)),insert_nat(A,B4)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ).

% insert_inter_insert
tff(fact_3997_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A4)),insert8211810215607154385at_nat(A,B4)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ).

% insert_inter_insert
tff(fact_3998_Int__insert__left__if1,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B4)),C2) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_3999_Int__insert__left__if1,axiom,
    ! [A: complex,C2: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(A),C2)
     => ( inf_inf_set_complex(insert_complex(A,B4),C2) = insert_complex(A,inf_inf_set_complex(B4,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4000_Int__insert__left__if1,axiom,
    ! [A: real,C2: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(A),C2)
     => ( inf_inf_set_real(insert_real(A,B4),C2) = insert_real(A,inf_inf_set_real(B4,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4001_Int__insert__left__if1,axiom,
    ! [A: $o,C2: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A)),C2)
     => ( inf_inf_set_o(insert_o((A),B4),C2) = insert_o((A),inf_inf_set_o(B4,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4002_Int__insert__left__if1,axiom,
    ! [A: int,C2: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A),C2)
     => ( inf_inf_set_int(insert_int(A,B4),C2) = insert_int(A,inf_inf_set_int(B4,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4003_Int__insert__left__if1,axiom,
    ! [A: nat,C2: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A),C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B4)),C2) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4004_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B4)),C2) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4005_Int__insert__left__if0,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B4)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2) ) ) ).

% Int_insert_left_if0
tff(fact_4006_Int__insert__left__if0,axiom,
    ! [A: complex,C2: set_complex,B4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A),C2)
     => ( inf_inf_set_complex(insert_complex(A,B4),C2) = inf_inf_set_complex(B4,C2) ) ) ).

% Int_insert_left_if0
tff(fact_4007_Int__insert__left__if0,axiom,
    ! [A: real,C2: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(A),C2)
     => ( inf_inf_set_real(insert_real(A,B4),C2) = inf_inf_set_real(B4,C2) ) ) ).

% Int_insert_left_if0
tff(fact_4008_Int__insert__left__if0,axiom,
    ! [A: $o,C2: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),C2)
     => ( inf_inf_set_o(insert_o((A),B4),C2) = inf_inf_set_o(B4,C2) ) ) ).

% Int_insert_left_if0
tff(fact_4009_Int__insert__left__if0,axiom,
    ! [A: int,C2: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),C2)
     => ( inf_inf_set_int(insert_int(A,B4),C2) = inf_inf_set_int(B4,C2) ) ) ).

% Int_insert_left_if0
tff(fact_4010_Int__insert__left__if0,axiom,
    ! [A: nat,C2: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2) ) ) ).

% Int_insert_left_if0
tff(fact_4011_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B4)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2) ) ) ).

% Int_insert_left_if0
tff(fact_4012_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S2),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_4013_Int__Un__eq_I4_J,axiom,
    ! [T2: set_nat,S2: 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,S2),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_4014_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,T2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S2),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_4015_Int__Un__eq_I3_J,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S2),T2)) = S2 ).

% Int_Un_eq(3)
tff(fact_4016_Int__Un__eq_I3_J,axiom,
    ! [S2: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S2),T2)) = S2 ).

% Int_Un_eq(3)
tff(fact_4017_Int__Un__eq_I3_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S2),T2)) = S2 ).

% Int_Un_eq(3)
tff(fact_4018_Int__Un__eq_I2_J,axiom,
    ! [S2: 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,S2),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_4019_Int__Un__eq_I2_J,axiom,
    ! [S2: 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,S2),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_4020_Int__Un__eq_I2_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S2),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_4021_Int__Un__eq_I1_J,axiom,
    ! [S2: 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,S2),T2)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_4022_Int__Un__eq_I1_J,axiom,
    ! [S2: 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,S2),T2)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_4023_Int__Un__eq_I1_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S2),T2)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_4024_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S2),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_4025_Un__Int__eq_I4_J,axiom,
    ! [T2: set_nat,S2: 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,S2),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_4026_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,T2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S2),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_4027_Un__Int__eq_I3_J,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S2),T2)) = S2 ).

% Un_Int_eq(3)
tff(fact_4028_Un__Int__eq_I3_J,axiom,
    ! [S2: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S2),T2)) = S2 ).

% Un_Int_eq(3)
tff(fact_4029_Un__Int__eq_I3_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S2),T2)) = S2 ).

% Un_Int_eq(3)
tff(fact_4030_Un__Int__eq_I2_J,axiom,
    ! [S2: 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,S2),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_4031_Un__Int__eq_I2_J,axiom,
    ! [S2: 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,S2),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_4032_Un__Int__eq_I2_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S2),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_4033_Un__Int__eq_I1_J,axiom,
    ! [S2: 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,S2),T2)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_4034_Un__Int__eq_I1_J,axiom,
    ! [S2: 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,S2),T2)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_4035_Un__Int__eq_I1_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S2),T2)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_4036_insert__disjoint_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A4)),B4) = bot_bo228742789529271731at_nat )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4037_insert__disjoint_I1_J,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( inf_inf_set_complex(insert_complex(A,A4),B4) = bot_bot_set_complex )
    <=> ( ~ aa_set_complex_o(member_complex(A),B4)
        & ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex ) ) ) ).

% insert_disjoint(1)
tff(fact_4038_insert__disjoint_I1_J,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( inf_inf_set_real(insert_real(A,A4),B4) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B4)
        & ( inf_inf_set_real(A4,B4) = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
tff(fact_4039_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A4)),B4) = bot_bo2099793752762293965at_nat )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4040_insert__disjoint_I1_J,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(insert_o((A),A4),B4) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B4)
        & ( inf_inf_set_o(A4,B4) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_4041_insert__disjoint_I1_J,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A4)),B4) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4042_insert__disjoint_I1_J,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int(insert_int(A,A4),B4) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B4)
        & ( inf_inf_set_int(A4,B4) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_4043_insert__disjoint_I2_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A4)),B4) )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4044_insert__disjoint_I2_J,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( bot_bot_set_complex = inf_inf_set_complex(insert_complex(A,A4),B4) )
    <=> ( ~ aa_set_complex_o(member_complex(A),B4)
        & ( bot_bot_set_complex = inf_inf_set_complex(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4045_insert__disjoint_I2_J,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(insert_real(A,A4),B4) )
    <=> ( ~ aa_set_real_o(member_real(A),B4)
        & ( bot_bot_set_real = inf_inf_set_real(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4046_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A4)),B4) )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4047_insert__disjoint_I2_J,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(insert_o((A),A4),B4) )
    <=> ( ~ aa_set_o_o(member_o((A)),B4)
        & ( bot_bot_set_o = inf_inf_set_o(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4048_insert__disjoint_I2_J,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A4)),B4) )
    <=> ( ~ aa_set_nat_o(member_nat(A),B4)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4049_insert__disjoint_I2_J,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(insert_int(A,A4),B4) )
    <=> ( ~ aa_set_int_o(member_int(A),B4)
        & ( bot_bot_set_int = inf_inf_set_int(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_4050_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),insert9069300056098147895at_nat(A,A4)) = bot_bo228742789529271731at_nat )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),A4) = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4051_disjoint__insert_I1_J,axiom,
    ! [B4: set_complex,A: complex,A4: set_complex] :
      ( ( inf_inf_set_complex(B4,insert_complex(A,A4)) = bot_bot_set_complex )
    <=> ( ~ aa_set_complex_o(member_complex(A),B4)
        & ( inf_inf_set_complex(B4,A4) = bot_bot_set_complex ) ) ) ).

% disjoint_insert(1)
tff(fact_4052_disjoint__insert_I1_J,axiom,
    ! [B4: set_real,A: real,A4: set_real] :
      ( ( inf_inf_set_real(B4,insert_real(A,A4)) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B4)
        & ( inf_inf_set_real(B4,A4) = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
tff(fact_4053_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),insert8211810215607154385at_nat(A,A4)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),A4) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4054_disjoint__insert_I1_J,axiom,
    ! [B4: set_o,A: $o,A4: set_o] :
      ( ( inf_inf_set_o(B4,insert_o((A),A4)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B4)
        & ( inf_inf_set_o(B4,A4) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_4055_disjoint__insert_I1_J,axiom,
    ! [B4: set_nat,A: nat,A4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),insert_nat(A,A4)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A4) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4056_disjoint__insert_I1_J,axiom,
    ! [B4: set_int,A: int,A4: set_int] :
      ( ( inf_inf_set_int(B4,insert_int(A,A4)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B4)
        & ( inf_inf_set_int(B4,A4) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_4057_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(B,B4)) )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A4)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4058_disjoint__insert_I2_J,axiom,
    ! [A4: set_complex,B: complex,B4: set_complex] :
      ( ( bot_bot_set_complex = inf_inf_set_complex(A4,insert_complex(B,B4)) )
    <=> ( ~ aa_set_complex_o(member_complex(B),A4)
        & ( bot_bot_set_complex = inf_inf_set_complex(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4059_disjoint__insert_I2_J,axiom,
    ! [A4: set_real,B: real,B4: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(A4,insert_real(B,B4)) )
    <=> ( ~ aa_set_real_o(member_real(B),A4)
        & ( bot_bot_set_real = inf_inf_set_real(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4060_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(B,B4)) )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A4)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4061_disjoint__insert_I2_J,axiom,
    ! [A4: set_o,B: $o,B4: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(A4,insert_o((B),B4)) )
    <=> ( ~ aa_set_o_o(member_o((B)),A4)
        & ( bot_bot_set_o = inf_inf_set_o(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4062_disjoint__insert_I2_J,axiom,
    ! [A4: set_nat,B: nat,B4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(B,B4)) )
    <=> ( ~ aa_set_nat_o(member_nat(B),A4)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4063_disjoint__insert_I2_J,axiom,
    ! [A4: set_int,B: int,B4: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(A4,insert_int(B,B4)) )
    <=> ( ~ aa_set_int_o(member_int(B),A4)
        & ( bot_bot_set_int = inf_inf_set_int(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_4064_Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_4065_Diff__disjoint,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(A4,minus_minus_set_o(B4,A4)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_4066_Diff__disjoint,axiom,
    ! [A4: set_int,B4: set_int] : inf_inf_set_int(A4,minus_minus_set_int(B4,A4)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_4067_Diff__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_4068_Collect__conj__eq,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] : collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dq(P),Q)) = inf_inf_set_complex(collect_complex(P),collect_complex(Q)) ).

% Collect_conj_eq
tff(fact_4069_Collect__conj__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dr(P),Q)) = inf_inf_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_conj_eq
tff(fact_4070_Collect__conj__eq,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ds(P),Q)) = inf_inf_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_conj_eq
tff(fact_4071_Collect__conj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dt(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_conj_eq
tff(fact_4072_Collect__conj__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] : collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_du(P),Q)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q)) ).

% Collect_conj_eq
tff(fact_4073_Int__Collect,axiom,
    ! [X3: real,A4: set_real,P: fun_real_o] :
      ( aa_set_real_o(member_real(X3),inf_inf_set_real(A4,collect_real(P)))
    <=> ( aa_set_real_o(member_real(X3),A4)
        & aa_real_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4074_Int__Collect,axiom,
    ! [X3: $o,A4: set_o,P: fun_o_o] :
      ( aa_set_o_o(member_o((X3)),inf_inf_set_o(A4,collect_o(P)))
    <=> ( aa_set_o_o(member_o((X3)),A4)
        & aa_o_o(P,(X3)) ) ) ).

% Int_Collect
tff(fact_4075_Int__Collect,axiom,
    ! [X3: int,A4: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(X3),inf_inf_set_int(A4,collect_int(P)))
    <=> ( aa_set_int_o(member_int(X3),A4)
        & aa_int_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4076_Int__Collect,axiom,
    ! [X3: complex,A4: set_complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(A4,collect_complex(P)))
    <=> ( aa_set_complex_o(member_complex(X3),A4)
        & aa_complex_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4077_Int__Collect,axiom,
    ! [X3: set_nat,A4: set_set_nat,P: fun_set_nat_o] :
      ( member_set_nat(X3,inf_inf_set_set_nat(A4,collect_set_nat(P)))
    <=> ( member_set_nat(X3,A4)
        & aa_set_nat_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4078_Int__Collect,axiom,
    ! [X3: list_nat,A4: set_list_nat,P: fun_list_nat_o] :
      ( member_list_nat(X3,inf_inf_set_list_nat(A4,collect_list_nat(P)))
    <=> ( member_list_nat(X3,A4)
        & aa_list_nat_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4079_Int__Collect,axiom,
    ! [X3: nat,A4: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P)))
    <=> ( aa_set_nat_o(member_nat(X3),A4)
        & aa_nat_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4080_Int__Collect,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),collec3392354462482085612at_nat(P)))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
        & aa_Pro6150637604445964947_nat_o(P,X3) ) ) ).

% Int_Collect
tff(fact_4081_Int__def,axiom,
    ! [A4: set_real,B4: set_real] : inf_inf_set_real(A4,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_dv(A4),B4)) ).

% Int_def
tff(fact_4082_Int__def,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(A4,B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_dw(A4),B4)) ).

% Int_def
tff(fact_4083_Int__def,axiom,
    ! [A4: set_int,B4: set_int] : inf_inf_set_int(A4,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_dx(A4),B4)) ).

% Int_def
tff(fact_4084_Int__def,axiom,
    ! [A4: set_complex,B4: set_complex] : inf_inf_set_complex(A4,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_dy(A4),B4)) ).

% Int_def
tff(fact_4085_Int__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : inf_inf_set_set_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_dz(A4),B4)) ).

% Int_def
tff(fact_4086_Int__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : inf_inf_set_list_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ea(A4),B4)) ).

% Int_def
tff(fact_4087_Int__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_eb(A4),B4)) ).

% Int_def
tff(fact_4088_Int__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_ec(A4),B4)) ).

% Int_def
tff(fact_4089_IntE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A4,B4))
     => ~ ( aa_set_complex_o(member_complex(C),A4)
         => ~ aa_set_complex_o(member_complex(C),B4) ) ) ).

% IntE
tff(fact_4090_IntE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A4,B4))
     => ~ ( aa_set_real_o(member_real(C),A4)
         => ~ aa_set_real_o(member_real(C),B4) ) ) ).

% IntE
tff(fact_4091_IntE,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A4,B4))
     => ~ ( aa_set_o_o(member_o((C)),A4)
         => ~ aa_set_o_o(member_o((C)),B4) ) ) ).

% IntE
tff(fact_4092_IntE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A4,B4))
     => ~ ( aa_set_int_o(member_int(C),A4)
         => ~ aa_set_int_o(member_int(C),B4) ) ) ).

% IntE
tff(fact_4093_IntE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
     => ~ ( aa_set_nat_o(member_nat(C),A4)
         => ~ aa_set_nat_o(member_nat(C),B4) ) ) ).

% IntE
tff(fact_4094_IntE,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A4)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B4) ) ) ).

% IntE
tff(fact_4095_IntD1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A4,B4))
     => aa_set_complex_o(member_complex(C),A4) ) ).

% IntD1
tff(fact_4096_IntD1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A4,B4))
     => aa_set_real_o(member_real(C),A4) ) ).

% IntD1
tff(fact_4097_IntD1,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A4,B4))
     => aa_set_o_o(member_o((C)),A4) ) ).

% IntD1
tff(fact_4098_IntD1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A4,B4))
     => aa_set_int_o(member_int(C),A4) ) ).

% IntD1
tff(fact_4099_IntD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
     => aa_set_nat_o(member_nat(C),A4) ) ).

% IntD1
tff(fact_4100_IntD1,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A4) ) ).

% IntD1
tff(fact_4101_IntD2,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A4,B4))
     => aa_set_complex_o(member_complex(C),B4) ) ).

% IntD2
tff(fact_4102_IntD2,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A4,B4))
     => aa_set_real_o(member_real(C),B4) ) ).

% IntD2
tff(fact_4103_IntD2,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A4,B4))
     => aa_set_o_o(member_o((C)),B4) ) ).

% IntD2
tff(fact_4104_IntD2,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A4,B4))
     => aa_set_int_o(member_int(C),B4) ) ).

% IntD2
tff(fact_4105_IntD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
     => aa_set_nat_o(member_nat(C),B4) ) ).

% IntD2
tff(fact_4106_IntD2,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B4) ) ).

% IntD2
tff(fact_4107_Int__assoc,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) ).

% Int_assoc
tff(fact_4108_Int__assoc,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: 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,A4),B4)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) ).

% Int_assoc
tff(fact_4109_Int__absorb,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),A4) = A4 ).

% Int_absorb
tff(fact_4110_Int__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),A4) = A4 ).

% Int_absorb
tff(fact_4111_Int__commute,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A4) ).

% Int_commute
tff(fact_4112_Int__commute,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),A4) ).

% Int_commute
tff(fact_4113_Int__left__absorb,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ).

% Int_left_absorb
tff(fact_4114_Int__left__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ).

% Int_left_absorb
tff(fact_4115_Int__left__commute,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C2)) ).

% Int_left_commute
tff(fact_4116_Int__left__commute,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C2)) ).

% Int_left_commute
tff(fact_4117_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_4118_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_4119_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_4120_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_4121_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_4122_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_4123_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_4124_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_4125_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_4126_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_4127_inf_Oabsorb__iff2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4128_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4129_inf_Oabsorb__iff2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4130_inf_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4131_inf_Oabsorb__iff2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4132_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4133_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4134_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4135_inf_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4136_inf_Oabsorb__iff1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4137_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_4138_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_4139_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),B) ).

% inf.cobounded2
tff(fact_4140_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_4141_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),B) ).

% inf.cobounded2
tff(fact_4142_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_4143_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_4144_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),A) ).

% inf.cobounded1
tff(fact_4145_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_4146_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),A) ).

% inf.cobounded1
tff(fact_4147_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_4148_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_4149_inf_Oorder__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.order_iff
tff(fact_4150_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_4151_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_4152_inf__greatest,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Z)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_4153_inf__greatest,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(X3),Z)
       => aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_4154_inf__greatest,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_4155_inf__greatest,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_4156_inf__greatest,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ) ) ).

% inf_greatest
tff(fact_4157_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_4158_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_4159_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C)) ) ) ).

% inf.boundedI
tff(fact_4160_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_4161_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ) ) ).

% inf.boundedI
tff(fact_4162_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_4163_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% inf.boundedE
tff(fact_4164_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% inf.boundedE
tff(fact_4165_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_4166_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.boundedE
tff(fact_4167_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Y),X3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4168_inf__absorb2,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),X3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4169_inf__absorb2,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4170_inf__absorb2,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4171_inf__absorb2,axiom,
    ! [Y: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X3)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4172_inf__absorb1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y) = X3 ) ) ).

% inf_absorb1
tff(fact_4173_inf__absorb1,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y) = X3 ) ) ).

% inf_absorb1
tff(fact_4174_inf__absorb1,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y) = X3 ) ) ).

% inf_absorb1
tff(fact_4175_inf__absorb1,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y) = X3 ) ) ).

% inf_absorb1
tff(fact_4176_inf__absorb1,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y) = X3 ) ) ).

% inf_absorb1
tff(fact_4177_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4178_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4179_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4180_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4181_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4182_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4183_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4184_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4185_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4186_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4187_le__iff__inf,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y) = X3 ) ) ).

% le_iff_inf
tff(fact_4188_le__iff__inf,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y) = X3 ) ) ).

% le_iff_inf
tff(fact_4189_le__iff__inf,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y) = X3 ) ) ).

% le_iff_inf
tff(fact_4190_le__iff__inf,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y) = X3 ) ) ).

% le_iff_inf
tff(fact_4191_le__iff__inf,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y) = X3 ) ) ).

% le_iff_inf
tff(fact_4192_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X5),Y3)),X5)
     => ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X5),Y3)),Y3)
       => ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X5),Y3)
             => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X5),Z2)
               => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X5),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y3),Z2)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X3),Y) ) ) ) ) ).

% inf_unique
tff(fact_4193_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X3: set_nat,Y: set_nat] :
      ( ! [X5: set_nat,Y3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3)),X5)
     => ( ! [X5: set_nat,Y3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3)),Y3)
       => ( ! [X5: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(X5),Y3)
             => ( aa_set_nat_o(ord_less_eq_set_nat(X5),Z2)
               => aa_set_nat_o(ord_less_eq_set_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),Z2)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y) ) ) ) ) ).

% inf_unique
tff(fact_4194_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X3: rat,Y: rat] :
      ( ! [X5: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3)),X5)
     => ( ! [X5: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3)),Y3)
       => ( ! [X5: rat,Y3: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Y3)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Z2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),Z2)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y) ) ) ) ) ).

% inf_unique
tff(fact_4195_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X3: nat,Y: nat] :
      ( ! [X5: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3)),X5)
     => ( ! [X5: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3)),Y3)
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),Z2)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y) ) ) ) ) ).

% inf_unique
tff(fact_4196_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,X3: int,Y: int] :
      ( ! [X5: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X5),Y3)),X5)
     => ( ! [X5: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X5),Y3)),Y3)
       => ( ! [X5: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Y3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Z2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),aa_int_int(aa_int_fun_int_int(F,Y3),Z2)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y) = aa_int_int(aa_int_fun_int_int(F,X3),Y) ) ) ) ) ).

% inf_unique
tff(fact_4197_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B) ) ).

% inf.orderI
tff(fact_4198_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% inf.orderI
tff(fact_4199_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% inf.orderI
tff(fact_4200_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% inf.orderI
tff(fact_4201_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% inf.orderI
tff(fact_4202_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.orderE
tff(fact_4203_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_4204_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.orderE
tff(fact_4205_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_4206_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_4207_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),X3)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X3) ) ).

% le_infI2
tff(fact_4208_le__infI2,axiom,
    ! [B: set_nat,X3: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),X3)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X3) ) ).

% le_infI2
tff(fact_4209_le__infI2,axiom,
    ! [B: rat,X3: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X3) ) ).

% le_infI2
tff(fact_4210_le__infI2,axiom,
    ! [B: nat,X3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X3) ) ).

% le_infI2
tff(fact_4211_le__infI2,axiom,
    ! [B: int,X3: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X3) ) ).

% le_infI2
tff(fact_4212_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),X3)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X3) ) ).

% le_infI1
tff(fact_4213_le__infI1,axiom,
    ! [A: set_nat,X3: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),X3)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X3) ) ).

% le_infI1
tff(fact_4214_le__infI1,axiom,
    ! [A: rat,X3: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X3) ) ).

% le_infI1
tff(fact_4215_le__infI1,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X3) ) ).

% le_infI1
tff(fact_4216_le__infI1,axiom,
    ! [A: int,X3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X3) ) ).

% le_infI1
tff(fact_4217_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),D)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),D)) ) ) ).

% inf_mono
tff(fact_4218_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),D)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D)) ) ) ).

% inf_mono
tff(fact_4219_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C),D)) ) ) ).

% inf_mono
tff(fact_4220_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D)) ) ) ).

% inf_mono
tff(fact_4221_inf__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D)) ) ) ).

% inf_mono
tff(fact_4222_le__infI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),A)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),B)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% le_infI
tff(fact_4223_le__infI,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(X3),B)
       => aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% le_infI
tff(fact_4224_le__infI,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)) ) ) ).

% le_infI
tff(fact_4225_le__infI,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% le_infI
tff(fact_4226_le__infI,axiom,
    ! [X3: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% le_infI
tff(fact_4227_le__infE,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),A)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),B) ) ) ).

% le_infE
tff(fact_4228_le__infE,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(X3),A)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(X3),B) ) ) ).

% le_infE
tff(fact_4229_le__infE,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),B) ) ) ).

% le_infE
tff(fact_4230_le__infE,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),B) ) ) ).

% le_infE
tff(fact_4231_le__infE,axiom,
    ! [X3: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),B) ) ) ).

% le_infE
tff(fact_4232_inf__le2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)),Y) ).

% inf_le2
tff(fact_4233_inf__le2,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)),Y) ).

% inf_le2
tff(fact_4234_inf__le2,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y)),Y) ).

% inf_le2
tff(fact_4235_inf__le2,axiom,
    ! [X3: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y)),Y) ).

% inf_le2
tff(fact_4236_inf__le2,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y)),Y) ).

% inf_le2
tff(fact_4237_inf__le1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)),X3) ).

% inf_le1
tff(fact_4238_inf__le1,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)),X3) ).

% inf_le1
tff(fact_4239_inf__le1,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y)),X3) ).

% inf_le1
tff(fact_4240_inf__le1,axiom,
    ! [X3: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y)),X3) ).

% inf_le1
tff(fact_4241_inf__le1,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y)),X3) ).

% inf_le1
tff(fact_4242_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)),X3) ).

% inf_sup_ord(1)
tff(fact_4243_inf__sup__ord_I1_J,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)),X3) ).

% inf_sup_ord(1)
tff(fact_4244_inf__sup__ord_I1_J,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y)),X3) ).

% inf_sup_ord(1)
tff(fact_4245_inf__sup__ord_I1_J,axiom,
    ! [X3: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y)),X3) ).

% inf_sup_ord(1)
tff(fact_4246_inf__sup__ord_I1_J,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y)),X3) ).

% inf_sup_ord(1)
tff(fact_4247_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_4248_inf__sup__ord_I2_J,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_4249_inf__sup__ord_I2_J,axiom,
    ! [X3: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_4250_inf__sup__ord_I2_J,axiom,
    ! [X3: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_4251_inf__sup__ord_I2_J,axiom,
    ! [X3: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_4252_disjoint__iff__not__equal,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
         => ! [Xa3: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),B4)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4253_disjoint__iff__not__equal,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A4)
         => ! [Xa3: $o] :
              ( aa_set_o_o(member_o((Xa3)),B4)
             => ( (X4)
              <=> ~ (Xa3) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4254_disjoint__iff__not__equal,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),B4)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4255_disjoint__iff__not__equal,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A4,B4) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),B4)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4256_Int__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_4257_Int__empty__right,axiom,
    ! [A4: set_o] : inf_inf_set_o(A4,bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_4258_Int__empty__right,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_4259_Int__empty__right,axiom,
    ! [A4: set_int] : inf_inf_set_int(A4,bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_4260_Int__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B4) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_4261_Int__empty__left,axiom,
    ! [B4: set_o] : inf_inf_set_o(bot_bot_set_o,B4) = bot_bot_set_o ).

% Int_empty_left
tff(fact_4262_Int__empty__left,axiom,
    ! [B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B4) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_4263_Int__empty__left,axiom,
    ! [B4: set_int] : inf_inf_set_int(bot_bot_set_int,B4) = bot_bot_set_int ).

% Int_empty_left
tff(fact_4264_disjoint__iff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),A4)
         => ~ aa_set_complex_o(member_complex(X4),B4) ) ) ).

% disjoint_iff
tff(fact_4265_disjoint__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( inf_inf_set_real(A4,B4) = bot_bot_set_real )
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A4)
         => ~ aa_set_real_o(member_real(X4),B4) ) ) ).

% disjoint_iff
tff(fact_4266_disjoint__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B4) ) ) ).

% disjoint_iff
tff(fact_4267_disjoint__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A4)
         => ~ aa_set_o_o(member_o((X4)),B4) ) ) ).

% disjoint_iff
tff(fact_4268_disjoint__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ~ aa_set_nat_o(member_nat(X4),B4) ) ) ).

% disjoint_iff
tff(fact_4269_disjoint__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A4,B4) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => ~ aa_set_int_o(member_int(X4),B4) ) ) ).

% disjoint_iff
tff(fact_4270_Int__emptyI,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => ~ aa_set_complex_o(member_complex(X5),B4) )
     => ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex ) ) ).

% Int_emptyI
tff(fact_4271_Int__emptyI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => ~ aa_set_real_o(member_real(X5),B4) )
     => ( inf_inf_set_real(A4,B4) = bot_bot_set_real ) ) ).

% Int_emptyI
tff(fact_4272_Int__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A4)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),B4) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_4273_Int__emptyI,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => ~ aa_set_o_o(member_o((X5)),B4) )
     => ( inf_inf_set_o(A4,B4) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_4274_Int__emptyI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => ~ aa_set_nat_o(member_nat(X5),B4) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_4275_Int__emptyI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => ~ aa_set_int_o(member_int(X5),B4) )
     => ( inf_inf_set_int(A4,B4) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_4276_Int__Collect__mono,axiom,
    ! [A4: set_real,B4: set_real,P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),A4)
           => ( aa_real_o(P,X5)
             => aa_real_o(Q,X5) ) )
       => aa_set_real_o(ord_less_eq_set_real(inf_inf_set_real(A4,collect_real(P))),inf_inf_set_real(B4,collect_real(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4277_Int__Collect__mono,axiom,
    ! [A4: set_o,B4: set_o,P: fun_o_o,Q: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
           => ( aa_o_o(P,(X5))
             => aa_o_o(Q,(X5)) ) )
       => aa_set_o_o(ord_less_eq_set_o(inf_inf_set_o(A4,collect_o(P))),inf_inf_set_o(B4,collect_o(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4278_Int__Collect__mono,axiom,
    ! [A4: set_int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
           => ( aa_int_o(P,X5)
             => aa_int_o(Q,X5) ) )
       => aa_set_int_o(ord_less_eq_set_int(inf_inf_set_int(A4,collect_int(P))),inf_inf_set_int(B4,collect_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4279_Int__Collect__mono,axiom,
    ! [A4: set_complex,B4: set_complex,P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => ( aa_complex_o(P,X5)
             => aa_complex_o(Q,X5) ) )
       => aa_set_complex_o(ord_le211207098394363844omplex(inf_inf_set_complex(A4,collect_complex(P))),inf_inf_set_complex(B4,collect_complex(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4280_Int__Collect__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(A4,B4)
     => ( ! [X5: set_nat] :
            ( member_set_nat(X5,A4)
           => ( aa_set_nat_o(P,X5)
             => aa_set_nat_o(Q,X5) ) )
       => ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A4,collect_set_nat(P)),inf_inf_set_set_nat(B4,collect_set_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4281_Int__Collect__mono,axiom,
    ! [A4: set_list_nat,B4: set_list_nat,P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A4,B4)
     => ( ! [X5: list_nat] :
            ( member_list_nat(X5,A4)
           => ( aa_list_nat_o(P,X5)
             => aa_list_nat_o(Q,X5) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A4,collect_list_nat(P)),inf_inf_set_list_nat(B4,collect_list_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4282_Int__Collect__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( ! [X5: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A4)
           => ( aa_Pro6150637604445964947_nat_o(P,X5)
             => aa_Pro6150637604445964947_nat_o(Q,X5) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),collec3392354462482085612at_nat(P))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),collec3392354462482085612at_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4283_Int__Collect__mono,axiom,
    ! [A4: set_nat,B4: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => ( aa_nat_o(P,X5)
             => aa_nat_o(Q,X5) ) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4284_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B4)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ) ) ).

% Int_greatest
tff(fact_4285_Int__greatest,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C2),B4)
       => aa_set_nat_o(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ) ) ).

% Int_greatest
tff(fact_4286_Int__absorb2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = A4 ) ) ).

% Int_absorb2
tff(fact_4287_Int__absorb2,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = A4 ) ) ).

% Int_absorb2
tff(fact_4288_Int__absorb1,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_4289_Int__absorb1,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_4290_Int__lower2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),B4) ).

% Int_lower2
tff(fact_4291_Int__lower2,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),B4) ).

% Int_lower2
tff(fact_4292_Int__lower1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),A4) ).

% Int_lower1
tff(fact_4293_Int__lower1,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),A4) ).

% Int_lower1
tff(fact_4294_Int__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),C2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),D2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D2)) ) ) ).

% Int_mono
tff(fact_4295_Int__mono,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat,D2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),D2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D2)) ) ) ).

% Int_mono
tff(fact_4296_Int__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(A,B4)) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4),insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) ).

% Int_insert_right
tff(fact_4297_Int__insert__right,axiom,
    ! [A4: set_complex,A: complex,B4: set_complex] :
      inf_inf_set_complex(A4,insert_complex(A,B4)) = $ite(aa_set_complex_o(member_complex(A),A4),insert_complex(A,inf_inf_set_complex(A4,B4)),inf_inf_set_complex(A4,B4)) ).

% Int_insert_right
tff(fact_4298_Int__insert__right,axiom,
    ! [A4: set_real,A: real,B4: set_real] :
      inf_inf_set_real(A4,insert_real(A,B4)) = $ite(aa_set_real_o(member_real(A),A4),insert_real(A,inf_inf_set_real(A4,B4)),inf_inf_set_real(A4,B4)) ).

% Int_insert_right
tff(fact_4299_Int__insert__right,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      inf_inf_set_o(A4,insert_o((A),B4)) = $ite(aa_set_o_o(member_o((A)),A4),insert_o((A),inf_inf_set_o(A4,B4)),inf_inf_set_o(A4,B4)) ).

% Int_insert_right
tff(fact_4300_Int__insert__right,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      inf_inf_set_int(A4,insert_int(A,B4)) = $ite(aa_set_int_o(member_int(A),A4),insert_int(A,inf_inf_set_int(A4,B4)),inf_inf_set_int(A4,B4)) ).

% Int_insert_right
tff(fact_4301_Int__insert__right,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(A,B4)) = $ite(aa_set_nat_o(member_nat(A),A4),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ).

% Int_insert_right
tff(fact_4302_Int__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(A,B4)) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ).

% Int_insert_right
tff(fact_4303_Int__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B4)),C2) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C2),insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2)) ).

% Int_insert_left
tff(fact_4304_Int__insert__left,axiom,
    ! [A: complex,B4: set_complex,C2: set_complex] :
      inf_inf_set_complex(insert_complex(A,B4),C2) = $ite(aa_set_complex_o(member_complex(A),C2),insert_complex(A,inf_inf_set_complex(B4,C2)),inf_inf_set_complex(B4,C2)) ).

% Int_insert_left
tff(fact_4305_Int__insert__left,axiom,
    ! [A: real,B4: set_real,C2: set_real] :
      inf_inf_set_real(insert_real(A,B4),C2) = $ite(aa_set_real_o(member_real(A),C2),insert_real(A,inf_inf_set_real(B4,C2)),inf_inf_set_real(B4,C2)) ).

% Int_insert_left
tff(fact_4306_Int__insert__left,axiom,
    ! [A: $o,B4: set_o,C2: set_o] :
      inf_inf_set_o(insert_o((A),B4),C2) = $ite(aa_set_o_o(member_o((A)),C2),insert_o((A),inf_inf_set_o(B4,C2)),inf_inf_set_o(B4,C2)) ).

% Int_insert_left
tff(fact_4307_Int__insert__left,axiom,
    ! [A: int,B4: set_int,C2: set_int] :
      inf_inf_set_int(insert_int(A,B4),C2) = $ite(aa_set_int_o(member_int(A),C2),insert_int(A,inf_inf_set_int(B4,C2)),inf_inf_set_int(B4,C2)) ).

% Int_insert_left
tff(fact_4308_Int__insert__left,axiom,
    ! [A: nat,B4: set_nat,C2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B4)),C2) = $ite(aa_set_nat_o(member_nat(A),C2),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) ).

% Int_insert_left
tff(fact_4309_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B4)),C2) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) ).

% Int_insert_left
tff(fact_4310_Un__Int__crazy,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: 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,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A4)) = 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,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A4)) ).

% Un_Int_crazy
tff(fact_4311_Un__Int__crazy,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A4)) = 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,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A4)) ).

% Un_Int_crazy
tff(fact_4312_Un__Int__crazy,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A4)) ).

% Un_Int_crazy
tff(fact_4313_Int__Un__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C2)) ).

% Int_Un_distrib
tff(fact_4314_Int__Un__distrib,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C2)) ).

% Int_Un_distrib
tff(fact_4315_Int__Un__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),C2)) ).

% Int_Un_distrib
tff(fact_4316_Un__Int__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),C2)) ).

% Un_Int_distrib
tff(fact_4317_Un__Int__distrib,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),C2)) ).

% Un_Int_distrib
tff(fact_4318_Un__Int__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),C2)) ).

% Un_Int_distrib
tff(fact_4319_Int__Un__distrib2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C2)),A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),A4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A4)) ).

% Int_Un_distrib2
tff(fact_4320_Int__Un__distrib2,axiom,
    ! [B4: set_nat,C2: set_nat,A4: 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,B4),C2)),A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A4)) ).

% Int_Un_distrib2
tff(fact_4321_Int__Un__distrib2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)),A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),A4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A4)) ).

% Int_Un_distrib2
tff(fact_4322_Un__Int__distrib2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)),A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),A4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A4)) ).

% Un_Int_distrib2
tff(fact_4323_Un__Int__distrib2,axiom,
    ! [B4: set_nat,C2: set_nat,A4: 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,B4),C2)),A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A4)) ).

% Un_Int_distrib2
tff(fact_4324_Un__Int__distrib2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2)),A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),A4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A4)) ).

% Un_Int_distrib2
tff(fact_4325_Int__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),C2)) ).

% Int_Diff
tff(fact_4326_Int__Diff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),C2)) ).

% Int_Diff
tff(fact_4327_Diff__Int2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C2)),B4) ).

% Diff_Int2
tff(fact_4328_Diff__Int2,axiom,
    ! [A4: set_nat,C2: set_nat,B4: 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,A4),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C2)),B4) ).

% Diff_Int2
tff(fact_4329_Diff__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ).

% Diff_Diff_Int
tff(fact_4330_Diff__Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ).

% Diff_Diff_Int
tff(fact_4331_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),B4)) ).

% Diff_Int_distrib
tff(fact_4332_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B4)) ).

% Diff_Int_distrib
tff(fact_4333_Diff__Int__distrib2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) ).

% Diff_Int_distrib2
tff(fact_4334_Diff__Int__distrib2,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) ).

% Diff_Int_distrib2
tff(fact_4335_distrib__sup__le,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Z))) ).

% distrib_sup_le
tff(fact_4336_distrib__sup__le,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Z))) ).

% distrib_sup_le
tff(fact_4337_distrib__sup__le,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Z))) ).

% distrib_sup_le
tff(fact_4338_distrib__sup__le,axiom,
    ! [X3: rat,Y: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Z))) ).

% distrib_sup_le
tff(fact_4339_distrib__sup__le,axiom,
    ! [X3: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Z))) ).

% distrib_sup_le
tff(fact_4340_distrib__sup__le,axiom,
    ! [X3: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Z))) ).

% distrib_sup_le
tff(fact_4341_distrib__inf__le,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_4342_distrib__inf__le,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),Z))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_4343_distrib__inf__le,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_4344_distrib__inf__le,axiom,
    ! [X3: rat,Y: rat,Z: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z))) ).

% distrib_inf_le
tff(fact_4345_distrib__inf__le,axiom,
    ! [X3: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_4346_distrib__inf__le,axiom,
    ! [X3: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z))) ).

% distrib_inf_le
tff(fact_4347_Diff__triv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = A4 ) ) ).

% Diff_triv
tff(fact_4348_Diff__triv,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
     => ( minus_minus_set_o(A4,B4) = A4 ) ) ).

% Diff_triv
tff(fact_4349_Diff__triv,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A4,B4) = bot_bot_set_int )
     => ( minus_minus_set_int(A4,B4) = A4 ) ) ).

% Diff_triv
tff(fact_4350_Diff__triv,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = A4 ) ) ).

% Diff_triv
tff(fact_4351_Int__Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_4352_Int__Diff__disjoint,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(inf_inf_set_o(A4,B4),minus_minus_set_o(A4,B4)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_4353_Int__Diff__disjoint,axiom,
    ! [A4: set_int,B4: set_int] : inf_inf_set_int(inf_inf_set_int(A4,B4),minus_minus_set_int(A4,B4)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_4354_Int__Diff__disjoint,axiom,
    ! [A4: set_nat,B4: 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,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_4355_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C2)) )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A4) ) ).

% Un_Int_assoc_eq
tff(fact_4356_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) )
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A4) ) ).

% Un_Int_assoc_eq
tff(fact_4357_Un__Int__assoc__eq,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) )
    <=> aa_set_nat_o(ord_less_eq_set_nat(C2),A4) ) ).

% Un_Int_assoc_eq
tff(fact_4358_Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),C2)) ).

% Diff_Un
tff(fact_4359_Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),C2)) ).

% Diff_Un
tff(fact_4360_Diff__Un,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),C2)) ).

% Diff_Un
tff(fact_4361_Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),C2)) ).

% Diff_Int
tff(fact_4362_Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),C2)) ).

% Diff_Int
tff(fact_4363_Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),C2)) ).

% Diff_Int
tff(fact_4364_Int__Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4365_Int__Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4366_Int__Diff__Un,axiom,
    ! [A4: set_nat,B4: 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,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4367_Un__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4368_Un__Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4369_Un__Diff__Int,axiom,
    ! [A4: set_nat,B4: 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,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4370_vebt__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_vebt_member(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xa2))
               => ~ $ite(
                      Xa2 = zero_zero_nat,
                      (A3),
                      $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary2),Xa2))
                 => ~ $ite(
                        Xa2 = Mi,
                        $true,
                        $ite(
                          Xa2 = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa2),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4371_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X3),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
           => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xa2 = Mi )
                        | ( Xa2 = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),Xa2)) ) )
             => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xa2 = Mi )
                          | ( Xa2 = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2),Xa2)) ) )
               => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4372_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(Uv)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),Xa2))
                   => ( ( Xa2 = Mi )
                      | ( Xa2 = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2),Xa2))
                     => ( ( Xa2 = Mi )
                        | ( Xa2 = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2),Xa2))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4373_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi )
                    | ( Xa2 = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi )
                      | ( Xa2 = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2),Xa2))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4374_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = insert_o((C),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & ( (B)
        <=> (C) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4375_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4376_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4377_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4378_atLeastAtMost__singleton,axiom,
    ! [A: $o] : set_or8904488021354931149Most_o((A),(A)) = insert_o((A),bot_bot_set_o) ).

% atLeastAtMost_singleton
tff(fact_4379_atLeastAtMost__singleton,axiom,
    ! [A: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(A),A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_4380_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_4381_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_4382_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_o,(B)),(A))
     => ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_4383_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4384_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4385_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4386_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4387_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4388_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
          & aa_set_nat_o(ord_less_eq_set_nat(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4389_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4390_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4391_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4392_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4393_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4394_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4395_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4396_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4397_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4398_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4399_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4400_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4401_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_4402_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4403_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_4404_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_4405_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4406_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_4407_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_4408_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
          & ~ aa_set_nat_o(ord_less_eq_set_nat(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4409_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_4410_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_4411_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4412_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_4413_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_4414_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4415_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4416_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4417_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4418_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4419_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4420_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4421_inf__Int__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(inf_in484816107290947356_nat_o(aTP_Lamm_bd(R),aTP_Lamm_bd(S2)),X),Xa)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Xa)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,R),S2)) ) ).

% inf_Int_eq2
tff(fact_4422_inf__Int__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num,X: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(inf_in2913458483842359172_num_o(aTP_Lamm_az(R),aTP_Lamm_az(S2)),X),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa),inf_in305939755129712355um_num(R,S2)) ) ).

% inf_Int_eq2
tff(fact_4423_inf__Int__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num,X: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(inf_in1133183212291220558_num_o(aTP_Lamm_ba(R),aTP_Lamm_ba(S2)),X),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa),inf_in7510916698018314649at_num(R,S2)) ) ).

% inf_Int_eq2
tff(fact_4424_inf__Int__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int,X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(inf_in8908015913983109984_int_o(aTP_Lamm_bc(R),aTP_Lamm_bc(S2)),X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),inf_in2269163501485487111nt_int(R,S2)) ) ).

% inf_Int_eq2
tff(fact_4425_inf__Int__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aTP_Lamm_bb(R),aTP_Lamm_bb(S2)),X),Xa)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,R),S2)) ) ).

% inf_Int_eq2
tff(fact_4426_inf__Int__eq,axiom,
    ! [R: set_complex,S2: set_complex,X: complex] :
      ( aa_complex_o(inf_in8039933884210822705plex_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S2)),X)
    <=> aa_set_complex_o(member_complex(X),inf_inf_set_complex(R,S2)) ) ).

% inf_Int_eq
tff(fact_4427_inf__Int__eq,axiom,
    ! [R: set_real,S2: set_real,X: real] :
      ( aa_real_o(inf_inf_fun_real_o(aTP_Lamm_a(R),aTP_Lamm_a(S2)),X)
    <=> aa_set_real_o(member_real(X),inf_inf_set_real(R,S2)) ) ).

% inf_Int_eq
tff(fact_4428_inf__Int__eq,axiom,
    ! [R: set_o,S2: set_o,X: $o] :
      ( aa_o_o(inf_inf_fun_o_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S2)),(X))
    <=> aa_set_o_o(member_o((X)),inf_inf_set_o(R,S2)) ) ).

% inf_Int_eq
tff(fact_4429_inf__Int__eq,axiom,
    ! [R: set_int,S2: set_int,X: int] :
      ( aa_int_o(inf_inf_fun_int_o(aTP_Lamm_ab(R),aTP_Lamm_ab(S2)),X)
    <=> aa_set_int_o(member_int(X),inf_inf_set_int(R,S2)) ) ).

% inf_Int_eq
tff(fact_4430_inf__Int__eq,axiom,
    ! [R: set_nat,S2: set_nat,X: nat] :
      ( aa_nat_o(inf_inf_fun_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S2)),X)
    <=> aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,R),S2)) ) ).

% inf_Int_eq
tff(fact_4431_inf__Int__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S2)),X)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,R),S2)) ) ).

% inf_Int_eq
tff(fact_4432_inf__set__def,axiom,
    ! [A4: set_real,B4: set_real] : inf_inf_set_real(A4,B4) = collect_real(inf_inf_fun_real_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% inf_set_def
tff(fact_4433_inf__set__def,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(A4,B4) = collect_o(inf_inf_fun_o_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ).

% inf_set_def
tff(fact_4434_inf__set__def,axiom,
    ! [A4: set_int,B4: set_int] : inf_inf_set_int(A4,B4) = collect_int(inf_inf_fun_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% inf_set_def
tff(fact_4435_inf__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] : inf_inf_set_complex(A4,B4) = collect_complex(inf_in8039933884210822705plex_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% inf_set_def
tff(fact_4436_inf__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : inf_inf_set_set_nat(A4,B4) = collect_set_nat(inf_in7602907407060649817_nat_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4))) ).

% inf_set_def
tff(fact_4437_inf__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : inf_inf_set_list_nat(A4,B4) = collect_list_nat(inf_in1107944134336960767_nat_o(aTP_Lamm_ag(A4),aTP_Lamm_ag(B4))) ).

% inf_set_def
tff(fact_4438_inf__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = collect_nat(inf_inf_fun_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% inf_set_def
tff(fact_4439_inf__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = collec3392354462482085612at_nat(inf_in3341167027036195756_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% inf_set_def
tff(fact_4440_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,X3: nat,M6: nat] :
      ( aa_nat_o(P,X3)
     => ( ! [X5: nat] :
            ( aa_nat_o(P,X5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M6) )
       => ~ ! [M: nat] :
              ( aa_nat_o(P,M)
             => ~ ! [X: nat] :
                    ( aa_nat_o(P,X)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_4441_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc8114805097830739791at_num] :
      ~ ! [F2: fun_nat_fun_num_num,A3: nat,B3: nat,Acc: num] : X3 != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A3,product_Pair_nat_num(B3,Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4442_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X3 != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4443_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,Mb: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),Mb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Mb),U)
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,Mb),set_or633870826150836451st_rat(Mb,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4444_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,Mb: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),Mb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,Mb),set_or7049704709247886629st_num(Mb,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4445_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,Mb: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4446_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,Mb: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),U)
       => ( sup_sup_set_int(set_or1266510415728281911st_int(L,Mb),set_or1266510415728281911st_int(Mb,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4447_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,Mb: real,U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Mb),U)
       => ( sup_sup_set_real(set_or1222579329274155063t_real(L,Mb),set_or1222579329274155063t_real(Mb,U)) = set_or1222579329274155063t_real(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4448_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( (A)
      <=> (B) )
     => ( set_or8904488021354931149Most_o((A),(B)) = insert_o((A),bot_bot_set_o) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4449_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4450_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( set_or1266510415728281911st_int(A,B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4451_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( set_or1222579329274155063t_real(A,B) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4452_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
            & aa_set_nat_o(ord_less_eq_set_nat(B),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A)
              | aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4453_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4454_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4455_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4456_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4457_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4458_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_4459_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) = insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4460_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4461_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_4462_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_4463_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_o] : inf_inf_set_o(X3,bot_bot_set_o) = bot_bot_set_o ).

% boolean_algebra.conj_zero_right
tff(fact_4464_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_4465_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_int] : inf_inf_set_int(X3,bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_4466_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X3) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_4467_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_o] : inf_inf_set_o(bot_bot_set_o,X3) = bot_bot_set_o ).

% boolean_algebra.conj_zero_left
tff(fact_4468_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X3) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_4469_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_int] : inf_inf_set_int(bot_bot_set_int,X3) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_4470_set__union,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : set_VEBT_VEBT2(union_VEBT_VEBT(Xs,Ys)) = sup_su6272177626956685416T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(Ys)) ).

% set_union
tff(fact_4471_set__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] : set_nat2(union_nat(Xs,Ys)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_nat2(Xs)),set_nat2(Ys)) ).

% set_union
tff(fact_4472_set__union,axiom,
    ! [Xs: list_P5464809261938338413at_nat,Ys: list_P5464809261938338413at_nat] : set_Pr3765526544606949372at_nat(union_4462254032241401953at_nat(Xs,Ys)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,set_Pr3765526544606949372at_nat(Xs)),set_Pr3765526544606949372at_nat(Ys)) ).

% set_union
tff(fact_4473_mult__le__cancel__iff1,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(X3),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4474_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4475_mult__le__cancel__iff1,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4476_mult__le__cancel__iff2,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),X3)),aa_real_real(times_times_real(Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4477_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4478_mult__le__cancel__iff2,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4479_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_4480_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_4481_low__def,axiom,
    ! [X3: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_4482_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4483_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_4484_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_4485_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_4486_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_4487_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_4488_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_4489_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_4490_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_4491_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_4492_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_4493_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_4494_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_4495_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_4496_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_4497_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_4498_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_4499_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_4500_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_4501_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_4502_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_4503_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_4504_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K2: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_4505_Suc__mod__mult__self3,axiom,
    ! [K2: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_4506_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_4507_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_4508_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_4509_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_4510_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_4511_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_4512_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_4513_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_4514_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_4515_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),Nb))),aa_num_nat(numeral_numeral_nat,K2)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_4516_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_4517_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_4518_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4519_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_4520_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_4521_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4522_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_4523_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_4524_mod2__gr__0,axiom,
    ! [Mb: 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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4525_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_4526_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_4527_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_4528_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_4529_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_4530_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_4531_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) ) ) ) ).

% mod_add_cong
tff(fact_4532_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ) ) ) ).

% mod_add_cong
tff(fact_4533_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A2,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B2,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),C) ) ) ) ).

% mod_add_cong
tff(fact_4534_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_4535_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_4536_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_4537_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_4538_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Nb)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),Nb)),B) ).

% power_mod
tff(fact_4539_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Nb),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),B) ).

% power_mod
tff(fact_4540_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_4541_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_4542_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_4543_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4544_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4545_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4546_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4547_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4548_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4549_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4550_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4551_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,one)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4552_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D4: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D4)) ) ).

% mod_eqE
tff(fact_4553_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D4: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D4)) ) ).

% mod_eqE
tff(fact_4554_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_4555_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_4556_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_4557_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_4558_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_4559_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_4560_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_4561_nat__mod__eq__iff,axiom,
    ! [X3: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_4562_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4563_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4564_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4565_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4566_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4567_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4568_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4569_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4570_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4571_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_4572_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_4573_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_4574_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_4575_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: 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,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_4576_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_4577_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_4578_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q3: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != 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,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_4579_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q3: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_4580_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4581_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4582_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_4583_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4584_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4585_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_4586_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_4587_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_4588_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_4589_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4590_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4591_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4592_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4593_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4594_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4595_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4596_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4597_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4598_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4599_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4600_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4601_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_4602_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_4603_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_4604_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_4605_nat__mod__eq__lemma,axiom,
    ! [X3: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
       => ? [Q2: nat] : X3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4606_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).

% mod_eq_nat2E
tff(fact_4607_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ~ ! [S: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).

% mod_eq_nat1E
tff(fact_4608_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_4609_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I4)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_4610_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4611_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4612_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4613_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( 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,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4614_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4615_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4616_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4617_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_4618_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_4619_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_4620_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_4621_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_4622_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,Mb: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_4623_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4624_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4625_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4626_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( 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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,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,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4627_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( 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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(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,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4628_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4629_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),bot_bo228742789529271731at_nat) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4630_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),bot_bo2099793752762293965at_nat) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4631_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_o] : sup_sup_set_o(X3,bot_bot_set_o) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4632_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),bot_bot_set_nat) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4633_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_int] : sup_sup_set_int(X3,bot_bot_set_int) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4634_mod__double__modulus,axiom,
    ! [Mb: code_integer,X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
       => ( ( modulo364778990260209775nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = modulo364778990260209775nteger(X3,Mb) )
          | ( modulo364778990260209775nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X3,Mb),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4635_mod__double__modulus,axiom,
    ! [Mb: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X3)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4636_mod__double__modulus,axiom,
    ! [Mb: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),Mb) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4637_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4638_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4639_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4640_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4641_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4642_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4643_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4644_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4645_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4646_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4647_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4648_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4649_diff__shunt__var,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X3),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y) ) ).

% diff_shunt_var
tff(fact_4650_diff__shunt__var,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( minus_minus_set_o(X3,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(X3),Y) ) ).

% diff_shunt_var
tff(fact_4651_diff__shunt__var,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X3,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(X3),Y) ) ).

% diff_shunt_var
tff(fact_4652_diff__shunt__var,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X3),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(X3),Y) ) ).

% diff_shunt_var
tff(fact_4653_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_4654_div__mod__decomp,axiom,
    ! [A4: nat,Nb: nat] : A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),Nb)) ).

% div_mod_decomp
tff(fact_4655_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4656_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4657_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4658_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_4659_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4660_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_4661_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4662_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4663_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4664_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Nb) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_4665_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Nb) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4666_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4667_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr8326237132889035090at_num(product_nat_num(Xs,Ys),Nb) = product_Pair_nat_num(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_4668_obtain__set__succ,axiom,
    ! [X3: nat,Z: nat,A4: set_nat,B4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A4),Z)
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( A4 = B4 )
           => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A4),X3),X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_4669_obtain__set__pred,axiom,
    ! [Z: nat,X3: nat,A4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X3)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A4),Z)
       => ( aa_set_nat_o(finite_finite_nat,A4)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A4),X3),X_13) ) ) ) ).

% obtain_set_pred
tff(fact_4670_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),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_4671_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),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_4672_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_4673_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A) ) ) ) ).

% pred_none_empty
tff(fact_4674_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X) ) ) ) ).

% succ_none_empty
tff(fact_4675_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_4676_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_4677_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_4678_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_4679_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs)) ).

% List.finite_set
tff(fact_4680_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4681_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4682_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_4683_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4684_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4685_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4686_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_4687_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4688_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4689_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4690_zmod__numeral__Bit0,axiom,
    ! [V2: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_4691_eucl__rel__int,axiom,
    ! [K2: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))) ).

% eucl_rel_int
tff(fact_4692_mod__int__unique,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = R2 ) ) ).

% mod_int_unique
tff(fact_4693_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4))
       => ( Q3 = Q5 ) ) ) ).

% unique_quotient
tff(fact_4694_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_4695_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M4: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M4) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_4696_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A4)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A4 ) ).

% finite_list
tff(fact_4697_finite__list,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A4 ) ).

% finite_list
tff(fact_4698_finite__list,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A4 ) ).

% finite_list
tff(fact_4699_finite__list,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A4 ) ).

% finite_list
tff(fact_4700_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed(F),U))) ) ).

% finite_less_ub
tff(fact_4701_finite__lists__length__eq,axiom,
    ! [A4: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_ee(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4702_finite__lists__length__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_ef(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4703_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_eg(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4704_finite__lists__length__eq,axiom,
    ! [A4: set_o,Nb: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_eh(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4705_finite__lists__length__eq,axiom,
    ! [A4: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_ei(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4706_finite__lists__length__eq,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ej(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4707_eucl__rel__int__by0,axiom,
    ! [K2: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K2)) ).

% eucl_rel_int_by0
tff(fact_4708_div__int__unique,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = Q3 ) ) ).

% div_int_unique
tff(fact_4709_finite__lists__length__le,axiom,
    ! [A4: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_ek(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4710_finite__lists__length__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_el(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4711_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_em(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4712_finite__lists__length__le,axiom,
    ! [A4: set_o,Nb: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_en(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4713_finite__lists__length__le,axiom,
    ! [A4: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_eo(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4714_finite__lists__length__le,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ep(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_4715_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4716_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_4717_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_4718_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_4719_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_4720_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_4721_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_4722_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_4723_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_4724_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_4725_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4726_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_4727_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_4728_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_4729_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_4730_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_4731_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_4732_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_4733_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_4734_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_4735_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_4736_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),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) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4737_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_4738_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_4739_max__def__raw,axiom,
    ! [X: set_nat,Xa: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4740_max__def__raw,axiom,
    ! [X: rat,Xa: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4741_max__def__raw,axiom,
    ! [X: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4742_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4743_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4744_finite__Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,insert_int(A,B4)))
    <=> aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_4745_finite__Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,minus_minus_set_o(A4,insert_o((A),B4)))
    <=> aa_set_o_o(finite_finite_o,minus_minus_set_o(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_4746_finite__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,B4)))
    <=> aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)) ) ).

% finite_Diff_insert
tff(fact_4747_finite__Diff__insert,axiom,
    ! [A4: set_complex,A: complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,insert_complex(A,B4)))
    <=> aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_4748_finite__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,B4)))
    <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) ) ).

% finite_Diff_insert
tff(fact_4749_finite__Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,B4)))
    <=> aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) ) ).

% finite_Diff_insert
tff(fact_4750_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eq,K2))) ).

% finite_Collect_le_nat
tff(fact_4751_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_er(A4))) ) ).

% finite_Collect_subsets
tff(fact_4752_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_es,A4))) ) ).

% finite_Collect_subsets
tff(fact_4753_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_et(A4))) ) ).

% finite_Collect_subsets
tff(fact_4754_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_real_o(finite_finite_real,collect_real(aTP_Lamm_eu(Nb))) ) ).

% finite_roots_unity
tff(fact_4755_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ev(Nb))) ) ).

% finite_roots_unity
tff(fact_4756_finite__induct__select,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S2)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [T3: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(T3),S2)
             => ( aa_set5844364881195198285_nat_o(P,T3)
               => ? [X: produc3843707927480180839at_nat] :
                    ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S2),T3))
                    & aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X,T3)) ) ) )
         => aa_set5844364881195198285_nat_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4757_finite__induct__select,axiom,
    ! [S2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T3: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T3),S2)
             => ( aa_set_complex_o(P,T3)
               => ? [X: complex] :
                    ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(S2,T3))
                    & aa_set_complex_o(P,insert_complex(X,T3)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4758_finite__induct__select,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [T3: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,T3),S2)
             => ( aa_set3702741264012733747_nat_o(P,T3)
               => ? [X: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S2),T3))
                    & aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X,T3)) ) ) )
         => aa_set3702741264012733747_nat_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4759_finite__induct__select,axiom,
    ! [S2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [T3: set_o] :
              ( aa_set_o_o(ord_less_set_o(T3),S2)
             => ( aa_set_o_o(P,T3)
               => ? [X: $o] :
                    ( aa_set_o_o(member_o((X)),minus_minus_set_o(S2,T3))
                    & aa_set_o_o(P,insert_o((X),T3)) ) ) )
         => aa_set_o_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4760_finite__induct__select,axiom,
    ! [S2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T3: set_int] :
              ( aa_set_int_o(ord_less_set_int(T3),S2)
             => ( aa_set_int_o(P,T3)
               => ? [X: int] :
                    ( aa_set_int_o(member_int(X),minus_minus_set_int(S2,T3))
                    & aa_set_int_o(P,insert_int(X,T3)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4761_finite__induct__select,axiom,
    ! [S2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T3: set_nat] :
              ( aa_set_nat_o(ord_less_set_nat(T3),S2)
             => ( aa_set_nat_o(P,T3)
               => ? [X: nat] :
                    ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S2),T3))
                    & aa_set_nat_o(P,insert_nat(X,T3)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4762_remove__induct,axiom,
    ! [P: fun_se7788542033685991124_nat_o,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
     => ( ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
         => aa_set5844364881195198285_nat_o(P,B4) )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7)
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A7),B4)
                 => ( ! [X: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A7)
                       => aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(P,A7) ) ) ) )
         => aa_set5844364881195198285_nat_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4763_remove__induct,axiom,
    ! [P: fun_set_real_o,B4: set_real] :
      ( aa_set_real_o(P,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B4)
         => aa_set_real_o(P,B4) )
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B4)
                 => ( ! [X: real] :
                        ( aa_set_real_o(member_real(X),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4764_remove__induct,axiom,
    ! [P: fun_set_complex_o,B4: set_complex] :
      ( aa_set_complex_o(P,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B4)
         => aa_set_complex_o(P,B4) )
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B4)
                 => ( ! [X: complex] :
                        ( aa_set_complex_o(member_complex(X),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4765_remove__induct,axiom,
    ! [P: fun_se4782206882984618170_nat_o,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
     => ( ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
         => aa_set3702741264012733747_nat_o(P,B4) )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A7),B4)
                 => ( ! [X: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A7)
                       => aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(P,A7) ) ) ) )
         => aa_set3702741264012733747_nat_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4766_remove__induct,axiom,
    ! [P: fun_set_o_o,B4: set_o] :
      ( aa_set_o_o(P,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B4)
         => aa_set_o_o(P,B4) )
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B4)
                 => ( ! [X: $o] :
                        ( aa_set_o_o(member_o((X)),A7)
                       => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o))) )
                   => aa_set_o_o(P,A7) ) ) ) )
         => aa_set_o_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4767_remove__induct,axiom,
    ! [P: fun_set_int_o,B4: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B4)
         => aa_set_int_o(P,B4) )
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B4)
                 => ( ! [X: int] :
                        ( aa_set_int_o(member_int(X),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4768_remove__induct,axiom,
    ! [P: fun_set_nat_o,B4: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B4)
         => aa_set_nat_o(P,B4) )
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B4)
                 => ( ! [X: nat] :
                        ( aa_set_nat_o(member_nat(X),A7)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(X,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_4769_finite__remove__induct,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7)
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A7),B4)
                 => ( ! [X: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A7)
                       => aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(P,A7) ) ) ) )
         => aa_set5844364881195198285_nat_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4770_finite__remove__induct,axiom,
    ! [B4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B4)
                 => ( ! [X: real] :
                        ( aa_set_real_o(member_real(X),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4771_finite__remove__induct,axiom,
    ! [B4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B4)
                 => ( ! [X: complex] :
                        ( aa_set_complex_o(member_complex(X),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4772_finite__remove__induct,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A7),B4)
                 => ( ! [X: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A7)
                       => aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(P,A7) ) ) ) )
         => aa_set3702741264012733747_nat_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4773_finite__remove__induct,axiom,
    ! [B4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B4)
                 => ( ! [X: $o] :
                        ( aa_set_o_o(member_o((X)),A7)
                       => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o))) )
                   => aa_set_o_o(P,A7) ) ) ) )
         => aa_set_o_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4774_finite__remove__induct,axiom,
    ! [B4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B4)
                 => ( ! [X: int] :
                        ( aa_set_int_o(member_int(X),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4775_finite__remove__induct,axiom,
    ! [B4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B4)
                 => ( ! [X: nat] :
                        ( aa_set_nat_o(member_nat(X),A7)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(X,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_4776_finite__insert,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,insert_int(A,A4))
    <=> aa_set_int_o(finite_finite_int,A4) ) ).

% finite_insert
tff(fact_4777_finite__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(finite_finite_o,insert_o((A),A4))
    <=> aa_set_o_o(finite_finite_o,A4) ) ).

% finite_insert
tff(fact_4778_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A4))
    <=> aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4) ) ).

% finite_insert
tff(fact_4779_finite__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,insert_nat(A,A4))
    <=> aa_set_nat_o(finite_finite_nat,A4) ) ).

% finite_insert
tff(fact_4780_finite__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A4))
    <=> aa_set_complex_o(finite3207457112153483333omplex,A4) ) ).

% finite_insert
tff(fact_4781_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A4))
    <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ).

% finite_insert
tff(fact_4782_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X)),N) ) ) ).

% finite_maxlen
tff(fact_4783_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N: nat] :
        ! [X: list_o] :
          ( member_list_o(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X)),N) ) ) ).

% finite_maxlen
tff(fact_4784_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N: nat] :
        ! [X: list_nat] :
          ( member_list_nat(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X)),N) ) ) ).

% finite_maxlen
tff(fact_4785_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X)),N) ) ) ).

% finite_maxlen
tff(fact_4786_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(A),A4)
       => ? [X5: real] :
            ( aa_set_real_o(member_real(X5),A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X5)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4787_finite__has__maximal2,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((A)),A4)
       => ? [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(X5))
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(Xa))
                 => ( (X5)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4788_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X5: set_nat] :
            ( member_set_nat(X5,A4)
            & aa_set_nat_o(ord_less_eq_set_nat(A),X5)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4789_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => ? [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X5)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4790_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => ? [X5: num] :
            ( aa_set_num_o(member_num(X5),A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X5)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4791_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X5)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4792_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X5)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4793_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(A),A4)
       => ? [X5: real] :
            ( aa_set_real_o(member_real(X5),A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4794_finite__has__minimal2,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((A)),A4)
       => ? [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(A))
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X5))
                 => ( (X5)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4795_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X5: set_nat] :
            ( member_set_nat(X5,A4)
            & aa_set_nat_o(ord_less_eq_set_nat(X5),A)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4796_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => ? [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4797_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => ? [X5: num] :
            ( aa_set_num_o(member_num(X5),A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4798_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4799_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4800_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_4801_finite_OemptyI,axiom,
    aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat) ).

% finite.emptyI
tff(fact_4802_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_4803_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_4804_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_4805_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_4806_infinite__imp__nonempty,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
     => ( S2 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_4807_infinite__imp__nonempty,axiom,
    ! [S2: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S2)
     => ( S2 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_4808_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_4809_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_4810_rev__finite__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% rev_finite_subset
tff(fact_4811_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_4812_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_4813_infinite__super,axiom,
    ! [S2: set_complex,T2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T2) ) ) ).

% infinite_super
tff(fact_4814_infinite__super,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S2),T2)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T2) ) ) ).

% infinite_super
tff(fact_4815_infinite__super,axiom,
    ! [S2: set_nat,T2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_super
tff(fact_4816_finite__subset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% finite_subset
tff(fact_4817_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_subset
tff(fact_4818_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_subset
tff(fact_4819_finite_OinsertI,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_set_int_o(finite_finite_int,insert_int(A,A4)) ) ).

% finite.insertI
tff(fact_4820_finite_OinsertI,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => aa_set_o_o(finite_finite_o,insert_o((A),A4)) ) ).

% finite.insertI
tff(fact_4821_finite_OinsertI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A4)) ) ).

% finite.insertI
tff(fact_4822_finite_OinsertI,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_nat_o(finite_finite_nat,insert_nat(A,A4)) ) ).

% finite.insertI
tff(fact_4823_finite_OinsertI,axiom,
    ! [A4: set_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A4)) ) ).

% finite.insertI
tff(fact_4824_finite_OinsertI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A4)) ) ).

% finite.insertI
tff(fact_4825_finite__has__minimal,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ? [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X5))
                 => ( (X5)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4826_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( member_set_nat(X5,A4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4827_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),A4)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4828_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X5: num] :
            ( aa_set_num_o(member_num(X5),A4)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4829_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4830_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X5)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4831_finite__has__maximal,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ? [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(Xa))
                 => ( (X5)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4832_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( member_set_nat(X5,A4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4833_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),A4)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4834_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X5: num] :
            ( aa_set_num_o(member_num(X5),A4)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4835_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4836_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Xa)
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4837_infinite__finite__induct,axiom,
    ! [P: fun_se7788542033685991124_nat_o,A4: set_Pr4329608150637261639at_nat] :
      ( ! [A7: set_Pr4329608150637261639at_nat] :
          ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7)
         => aa_set5844364881195198285_nat_o(P,A7) )
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [X5: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
             => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),F3)
               => ( aa_set5844364881195198285_nat_o(P,F3)
                 => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X5,F3)) ) ) )
         => aa_set5844364881195198285_nat_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4838_infinite__finite__induct,axiom,
    ! [P: fun_set_real_o,A4: set_real] :
      ( ! [A7: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A7)
         => aa_set_real_o(P,A7) )
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X5: real,F3: set_real] :
              ( aa_set_real_o(finite_finite_real,F3)
             => ( ~ aa_set_real_o(member_real(X5),F3)
               => ( aa_set_real_o(P,F3)
                 => aa_set_real_o(P,insert_real(X5,F3)) ) ) )
         => aa_set_real_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4839_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_o,A4: set_complex] :
      ( ! [A7: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A7)
         => aa_set_complex_o(P,A7) )
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X5: complex,F3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F3)
             => ( ~ aa_set_complex_o(member_complex(X5),F3)
               => ( aa_set_complex_o(P,F3)
                 => aa_set_complex_o(P,insert_complex(X5,F3)) ) ) )
         => aa_set_complex_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4840_infinite__finite__induct,axiom,
    ! [P: fun_se4782206882984618170_nat_o,A4: set_Pr1261947904930325089at_nat] :
      ( ! [A7: set_Pr1261947904930325089at_nat] :
          ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
         => aa_set3702741264012733747_nat_o(P,A7) )
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [X5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
             => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),F3)
               => ( aa_set3702741264012733747_nat_o(P,F3)
                 => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X5,F3)) ) ) )
         => aa_set3702741264012733747_nat_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4841_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A4: set_o] :
      ( ! [A7: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A7)
         => aa_set_o_o(P,A7) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X5: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X5)),F3)
               => ( aa_set_o_o(P,F3)
                 => aa_set_o_o(P,insert_o((X5),F3)) ) ) )
         => aa_set_o_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4842_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A4: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A7)
         => aa_set_nat_o(P,A7) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X5: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X5),F3)
               => ( aa_set_nat_o(P,F3)
                 => aa_set_nat_o(P,insert_nat(X5,F3)) ) ) )
         => aa_set_nat_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4843_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A4: set_int] :
      ( ! [A7: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A7)
         => aa_set_int_o(P,A7) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X5: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X5),F3)
               => ( aa_set_int_o(P,F3)
                 => aa_set_int_o(P,insert_int(X5,F3)) ) ) )
         => aa_set_int_o(P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_4844_finite__ne__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( ( F4 != bot_bo228742789529271731at_nat )
       => ( ! [X5: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X5,bot_bo228742789529271731at_nat))
         => ( ! [X5: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
               => ( ( F3 != bot_bo228742789529271731at_nat )
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),F3)
                   => ( aa_set5844364881195198285_nat_o(P,F3)
                     => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X5,F3)) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4845_finite__ne__induct,axiom,
    ! [F4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F4)
     => ( ( F4 != bot_bot_set_real )
       => ( ! [X5: real] : aa_set_real_o(P,insert_real(X5,bot_bot_set_real))
         => ( ! [X5: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( ( F3 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real(X5),F3)
                   => ( aa_set_real_o(P,F3)
                     => aa_set_real_o(P,insert_real(X5,F3)) ) ) ) )
           => aa_set_real_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4846_finite__ne__induct,axiom,
    ! [F4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( ( F4 != bot_bot_set_complex )
       => ( ! [X5: complex] : aa_set_complex_o(P,insert_complex(X5,bot_bot_set_complex))
         => ( ! [X5: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( ( F3 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X5),F3)
                   => ( aa_set_complex_o(P,F3)
                     => aa_set_complex_o(P,insert_complex(X5,F3)) ) ) ) )
           => aa_set_complex_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4847_finite__ne__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( ( F4 != bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat))
         => ( ! [X5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
               => ( ( F3 != bot_bo2099793752762293965at_nat )
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),F3)
                   => ( aa_set3702741264012733747_nat_o(P,F3)
                     => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X5,F3)) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4848_finite__ne__induct,axiom,
    ! [F4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( ( F4 != bot_bot_set_o )
       => ( ! [X5: $o] : aa_set_o_o(P,insert_o((X5),bot_bot_set_o))
         => ( ! [X5: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X5)),F3)
                   => ( aa_set_o_o(P,F3)
                     => aa_set_o_o(P,insert_o((X5),F3)) ) ) ) )
           => aa_set_o_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4849_finite__ne__induct,axiom,
    ! [F4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( ( F4 != bot_bot_set_nat )
       => ( ! [X5: nat] : aa_set_nat_o(P,insert_nat(X5,bot_bot_set_nat))
         => ( ! [X5: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X5),F3)
                   => ( aa_set_nat_o(P,F3)
                     => aa_set_nat_o(P,insert_nat(X5,F3)) ) ) ) )
           => aa_set_nat_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4850_finite__ne__induct,axiom,
    ! [F4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( ( F4 != bot_bot_set_int )
       => ( ! [X5: int] : aa_set_int_o(P,insert_int(X5,bot_bot_set_int))
         => ( ! [X5: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X5),F3)
                   => ( aa_set_int_o(P,F3)
                     => aa_set_int_o(P,insert_int(X5,F3)) ) ) ) )
           => aa_set_int_o(P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_4851_finite__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [X5: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
             => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),F3)
               => ( aa_set5844364881195198285_nat_o(P,F3)
                 => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X5,F3)) ) ) )
         => aa_set5844364881195198285_nat_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4852_finite__induct,axiom,
    ! [F4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F4)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X5: real,F3: set_real] :
              ( aa_set_real_o(finite_finite_real,F3)
             => ( ~ aa_set_real_o(member_real(X5),F3)
               => ( aa_set_real_o(P,F3)
                 => aa_set_real_o(P,insert_real(X5,F3)) ) ) )
         => aa_set_real_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4853_finite__induct,axiom,
    ! [F4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X5: complex,F3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F3)
             => ( ~ aa_set_complex_o(member_complex(X5),F3)
               => ( aa_set_complex_o(P,F3)
                 => aa_set_complex_o(P,insert_complex(X5,F3)) ) ) )
         => aa_set_complex_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4854_finite__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [X5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
             => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),F3)
               => ( aa_set3702741264012733747_nat_o(P,F3)
                 => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X5,F3)) ) ) )
         => aa_set3702741264012733747_nat_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4855_finite__induct,axiom,
    ! [F4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X5: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X5)),F3)
               => ( aa_set_o_o(P,F3)
                 => aa_set_o_o(P,insert_o((X5),F3)) ) ) )
         => aa_set_o_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4856_finite__induct,axiom,
    ! [F4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X5: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X5),F3)
               => ( aa_set_nat_o(P,F3)
                 => aa_set_nat_o(P,insert_nat(X5,F3)) ) ) )
         => aa_set_nat_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4857_finite__induct,axiom,
    ! [F4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X5: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X5),F3)
               => ( aa_set_int_o(P,F3)
                 => aa_set_int_o(P,insert_int(X5,F3)) ) ) )
         => aa_set_int_o(P,F4) ) ) ) ).

% finite_induct
tff(fact_4858_finite_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
    <=> ( ( A = bot_bo228742789529271731at_nat )
        | ? [A8: set_Pr4329608150637261639at_nat,A6: produc3843707927480180839at_nat] :
            ( ( A = insert9069300056098147895at_nat(A6,A8) )
            & aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A8) ) ) ) ).

% finite.simps
tff(fact_4859_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
    <=> ( ( A = bot_bot_set_complex )
        | ? [A8: set_complex,A6: complex] :
            ( ( A = insert_complex(A6,A8) )
            & aa_set_complex_o(finite3207457112153483333omplex,A8) ) ) ) ).

% finite.simps
tff(fact_4860_finite_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        | ? [A8: set_Pr1261947904930325089at_nat,A6: product_prod_nat_nat] :
            ( ( A = insert8211810215607154385at_nat(A6,A8) )
            & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A8) ) ) ) ).

% finite.simps
tff(fact_4861_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A8: set_o,A6: $o] :
            ( ( A = insert_o((A6),A8) )
            & aa_set_o_o(finite_finite_o,A8) ) ) ) ).

% finite.simps
tff(fact_4862_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A8: set_nat,A6: nat] :
            ( ( A = insert_nat(A6,A8) )
            & aa_set_nat_o(finite_finite_nat,A8) ) ) ) ).

% finite.simps
tff(fact_4863_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A8: set_int,A6: int] :
            ( ( A = insert_int(A6,A8) )
            & aa_set_int_o(finite_finite_int,A8) ) ) ) ).

% finite.simps
tff(fact_4864_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A7: set_Pr4329608150637261639at_nat] :
              ( ? [A3: produc3843707927480180839at_nat] : A = insert9069300056098147895at_nat(A3,A7)
             => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7) ) ) ) ).

% finite.cases
tff(fact_4865_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A7: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A7)
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A7) ) ) ) ).

% finite.cases
tff(fact_4866_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A7: set_Pr1261947904930325089at_nat] :
              ( ? [A3: product_prod_nat_nat] : A = insert8211810215607154385at_nat(A3,A7)
             => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7) ) ) ) ).

% finite.cases
tff(fact_4867_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A3: $o] : A = insert_o((A3),A7)
             => ~ aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.cases
tff(fact_4868_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A3: nat] : A = insert_nat(A3,A7)
             => ~ aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.cases
tff(fact_4869_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A3: int] : A = insert_int(A3,A7)
             => ~ aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.cases
tff(fact_4870_finite__subset__induct_H,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),A4)
       => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
         => ( ! [A3: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A4)
                 => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),A4)
                   => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F3)
                     => ( aa_set5844364881195198285_nat_o(P,F3)
                       => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(A3,F3)) ) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4871_finite__subset__induct_H,axiom,
    ! [F4: set_real,A4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F4)
     => ( aa_set_real_o(ord_less_eq_set_real(F4),A4)
       => ( 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),A4)
                 => ( aa_set_real_o(ord_less_eq_set_real(F3),A4)
                   => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4872_finite__subset__induct_H,axiom,
    ! [F4: set_complex,A4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A4)
       => ( 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),A4)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A4)
                   => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4873_finite__subset__induct_H,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F4),A4)
       => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
         => ( ! [A3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A4)
                 => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F3),A4)
                   => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F3)
                     => ( aa_set3702741264012733747_nat_o(P,F3)
                       => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(A3,F3)) ) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4874_finite__subset__induct_H,axiom,
    ! [F4: set_o,A4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(F4),A4)
       => ( 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)),A4)
                 => ( aa_set_o_o(ord_less_eq_set_o(F3),A4)
                   => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4875_finite__subset__induct_H,axiom,
    ! [F4: set_int,A4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(ord_less_eq_set_int(F4),A4)
       => ( 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),A4)
                 => ( aa_set_int_o(ord_less_eq_set_int(F3),A4)
                   => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4876_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F4),A4)
       => ( 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),A4)
                 => ( aa_set_nat_o(ord_less_eq_set_nat(F3),A4)
                   => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4877_finite__subset__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),A4)
       => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
         => ( ! [A3: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A4)
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F3)
                   => ( aa_set5844364881195198285_nat_o(P,F3)
                     => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(A3,F3)) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4878_finite__subset__induct,axiom,
    ! [F4: set_real,A4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F4)
     => ( aa_set_real_o(ord_less_eq_set_real(F4),A4)
       => ( 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),A4)
                 => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4879_finite__subset__induct,axiom,
    ! [F4: set_complex,A4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A4)
       => ( 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),A4)
                 => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4880_finite__subset__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F4),A4)
       => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
         => ( ! [A3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A4)
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F3)
                   => ( aa_set3702741264012733747_nat_o(P,F3)
                     => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(A3,F3)) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4881_finite__subset__induct,axiom,
    ! [F4: set_o,A4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(F4),A4)
       => ( 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)),A4)
                 => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4882_finite__subset__induct,axiom,
    ! [F4: set_int,A4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(ord_less_eq_set_int(F4),A4)
       => ( 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),A4)
                 => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4883_finite__subset__induct,axiom,
    ! [F4: set_nat,A4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F4),A4)
       => ( 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),A4)
                 => ( ~ 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,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_4884_finite__empty__induct,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( aa_set5844364881195198285_nat_o(P,A4)
       => ( ! [A3: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7)
             => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A7)
               => ( aa_set5844364881195198285_nat_o(P,A7)
                 => aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(A3,bot_bo228742789529271731at_nat))) ) ) )
         => aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat) ) ) ) ).

% finite_empty_induct
tff(fact_4885_finite__empty__induct,axiom,
    ! [A4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(P,A4)
       => ( ! [A3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( aa_set_real_o(member_real(A3),A7)
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(A3,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_4886_finite__empty__induct,axiom,
    ! [A4: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(P,A4)
       => ( ! [A3: complex,A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( aa_set_complex_o(member_complex(A3),A7)
               => ( aa_set_complex_o(P,A7)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_4887_finite__empty__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(P,A4)
       => ( ! [A3: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
             => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A7)
               => ( aa_set3702741264012733747_nat_o(P,A7)
                 => aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(A3,bot_bo2099793752762293965at_nat))) ) ) )
         => aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat) ) ) ) ).

% finite_empty_induct
tff(fact_4888_finite__empty__induct,axiom,
    ! [A4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(P,A4)
       => ( ! [A3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( aa_set_o_o(member_o((A3)),A7)
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((A3),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_4889_finite__empty__induct,axiom,
    ! [A4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(P,A4)
       => ( ! [A3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( aa_set_int_o(member_int(A3),A7)
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(A3,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_4890_finite__empty__induct,axiom,
    ! [A4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(P,A4)
       => ( ! [A3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( aa_set_nat_o(member_nat(A3),A7)
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_4891_infinite__coinduct,axiom,
    ! [X6: fun_se7788542033685991124_nat_o,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(X6,A4)
     => ( ! [A7: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(X6,A7)
           => ? [X: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A7)
                & ( aa_set5844364881195198285_nat_o(X6,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)))
                  | ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))) ) ) )
       => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4) ) ) ).

% infinite_coinduct
tff(fact_4892_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_o,A4: set_complex] :
      ( aa_set_complex_o(X6,A4)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(X6,A7)
           => ? [X: complex] :
                ( aa_set_complex_o(member_complex(X),A7)
                & ( aa_set_complex_o(X6,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% infinite_coinduct
tff(fact_4893_infinite__coinduct,axiom,
    ! [X6: fun_se4782206882984618170_nat_o,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(X6,A4)
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(X6,A7)
           => ? [X: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A7)
                & ( aa_set3702741264012733747_nat_o(X6,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))
                  | ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) ) ) )
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% infinite_coinduct
tff(fact_4894_infinite__coinduct,axiom,
    ! [X6: fun_set_o_o,A4: set_o] :
      ( aa_set_o_o(X6,A4)
     => ( ! [A7: set_o] :
            ( aa_set_o_o(X6,A7)
           => ? [X: $o] :
                ( aa_set_o_o(member_o((X)),A7)
                & ( aa_set_o_o(X6,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A4) ) ) ).

% infinite_coinduct
tff(fact_4895_infinite__coinduct,axiom,
    ! [X6: fun_set_int_o,A4: set_int] :
      ( aa_set_int_o(X6,A4)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(X6,A7)
           => ? [X: int] :
                ( aa_set_int_o(member_int(X),A7)
                & ( aa_set_int_o(X6,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A4) ) ) ).

% infinite_coinduct
tff(fact_4896_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_o,A4: set_nat] :
      ( aa_set_nat_o(X6,A4)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(X6,A7)
           => ? [X: nat] :
                ( aa_set_nat_o(member_nat(X),A7)
                & ( aa_set_nat_o(X6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(X,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(X,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% infinite_coinduct
tff(fact_4897_infinite__remove,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S2)
     => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) ) ).

% infinite_remove
tff(fact_4898_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_4899_infinite__remove,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S2)
     => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) ) ).

% infinite_remove
tff(fact_4900_infinite__remove,axiom,
    ! [S2: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S2)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_4901_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_4902_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S2),insert_nat(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_4903_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ew(Nb),C))) ) ).

% finite_nth_roots
tff(fact_4904_set__encode__insert,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Nb),A4)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_4905_finite__linorder__min__induct,axiom,
    ! [A4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X: $o] :
                    ( aa_set_o_o(member_o((X)),A7)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B3)),(X)) )
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,insert_o((B3),A7)) ) ) )
         => aa_set_o_o(P,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4906_finite__linorder__min__induct,axiom,
    ! [A4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B3),X) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B3,A7)) ) ) )
         => aa_set_real_o(P,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4907_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X: rat] :
                    ( aa_set_rat_o(member_rat(X),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),X) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B3,A7)) ) ) )
         => aa_set_rat_o(P,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4908_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X: num] :
                    ( aa_set_num_o(member_num(X),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B3),X) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B3,A7)) ) ) )
         => aa_set_num_o(P,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4909_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X: nat] :
                    ( aa_set_nat_o(member_nat(X),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B3,A7)) ) ) )
         => aa_set_nat_o(P,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4910_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X: int] :
                    ( aa_set_int_o(member_int(X),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B3,A7)) ) ) )
         => aa_set_int_o(P,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4911_finite__linorder__max__induct,axiom,
    ! [A4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X: $o] :
                    ( aa_set_o_o(member_o((X)),A7)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(B3)) )
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,insert_o((B3),A7)) ) ) )
         => aa_set_o_o(P,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4912_finite__linorder__max__induct,axiom,
    ! [A4: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X),B3) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B3,A7)) ) ) )
         => aa_set_real_o(P,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4913_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X: rat] :
                    ( aa_set_rat_o(member_rat(X),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),B3) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B3,A7)) ) ) )
         => aa_set_rat_o(P,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4914_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X: num] :
                    ( aa_set_num_o(member_num(X),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X),B3) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B3,A7)) ) ) )
         => aa_set_num_o(P,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4915_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X: nat] :
                    ( aa_set_nat_o(member_nat(X),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B3) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B3,A7)) ) ) )
         => aa_set_nat_o(P,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4916_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X: int] :
                    ( aa_set_int_o(member_int(X),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X),B3) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B3,A7)) ) ) )
         => aa_set_int_o(P,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4917_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X5: real,S3: set_real] :
              ( aa_set_real_o(finite_finite_real,S3)
             => ( ! [Y5: real] :
                    ( aa_set_real_o(member_real(Y5),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y5)),aa_real_rat(F,X5)) )
               => ( aa_set_real_o(P,S3)
                 => aa_set_real_o(P,insert_real(X5,S3)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4918_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X5: complex,S3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S3)
             => ( ! [Y5: complex] :
                    ( aa_set_complex_o(member_complex(Y5),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y5)),aa_complex_rat(F,X5)) )
               => ( aa_set_complex_o(P,S3)
                 => aa_set_complex_o(P,insert_complex(X5,S3)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4919_finite__ranking__induct,axiom,
    ! [S2: set_o,P: fun_set_o_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X5: $o,S3: set_o] :
              ( aa_set_o_o(finite_finite_o,S3)
             => ( ! [Y5: $o] :
                    ( aa_set_o_o(member_o((Y5)),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Y5))),aa_o_rat(F,(X5))) )
               => ( aa_set_o_o(P,S3)
                 => aa_set_o_o(P,insert_o((X5),S3)) ) ) )
         => aa_set_o_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4920_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X5: nat,S3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S3)
             => ( ! [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y5)),aa_nat_rat(F,X5)) )
               => ( aa_set_nat_o(P,S3)
                 => aa_set_nat_o(P,insert_nat(X5,S3)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4921_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X5: int,S3: set_int] :
              ( aa_set_int_o(finite_finite_int,S3)
             => ( ! [Y5: int] :
                    ( aa_set_int_o(member_int(Y5),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y5)),aa_int_rat(F,X5)) )
               => ( aa_set_int_o(P,S3)
                 => aa_set_int_o(P,insert_int(X5,S3)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4922_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_o,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X5: real,S3: set_real] :
              ( aa_set_real_o(finite_finite_real,S3)
             => ( ! [Y5: real] :
                    ( aa_set_real_o(member_real(Y5),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y5)),aa_real_num(F,X5)) )
               => ( aa_set_real_o(P,S3)
                 => aa_set_real_o(P,insert_real(X5,S3)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4923_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X5: complex,S3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S3)
             => ( ! [Y5: complex] :
                    ( aa_set_complex_o(member_complex(Y5),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y5)),aa_complex_num(F,X5)) )
               => ( aa_set_complex_o(P,S3)
                 => aa_set_complex_o(P,insert_complex(X5,S3)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4924_finite__ranking__induct,axiom,
    ! [S2: set_o,P: fun_set_o_o,F: fun_o_num] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X5: $o,S3: set_o] :
              ( aa_set_o_o(finite_finite_o,S3)
             => ( ! [Y5: $o] :
                    ( aa_set_o_o(member_o((Y5)),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_o_num(F,(Y5))),aa_o_num(F,(X5))) )
               => ( aa_set_o_o(P,S3)
                 => aa_set_o_o(P,insert_o((X5),S3)) ) ) )
         => aa_set_o_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4925_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X5: nat,S3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S3)
             => ( ! [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y5)),aa_nat_num(F,X5)) )
               => ( aa_set_nat_o(P,S3)
                 => aa_set_nat_o(P,insert_nat(X5,S3)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4926_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_o,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X5: int,S3: set_int] :
              ( aa_set_int_o(finite_finite_int,S3)
             => ( ! [Y5: int] :
                    ( aa_set_int_o(member_int(Y5),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y5)),aa_int_num(F,X5)) )
               => ( aa_set_int_o(P,S3)
                 => aa_set_int_o(P,insert_int(X5,S3)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_4927_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_ex(I5),X3)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_ex(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ey(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4928_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X3: fun_o_real,Y: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_ez(I5),X3)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_ez(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aa_fun3246320876514483410un_o_o(aTP_Lamm_fa(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4929_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fb(I5),X3)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fb(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_fc(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4930_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_fd(I5),X3)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_fd(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_fe(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4931_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ff(I5),X3)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ff(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_fg(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4932_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fh(I5),X3)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fh(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fi(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4933_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X3: fun_o_rat,Y: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_fj(I5),X3)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_fj(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_fk(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4934_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_fl(I5),X3)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_fl(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_fm(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4935_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_fn(I5),X3)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_fn(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_fo(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4936_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: fun_complex_rat,Y: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_fp(I5),X3)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_fp(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_fq(I5),X3),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4937_concat__bit__Suc,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Nb)),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_4938_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_4939_concat__bit__assoc,axiom,
    ! [Nb: nat,K2: int,Mb: nat,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L)),R2) ).

% concat_bit_assoc
tff(fact_4940_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_nat(nat_set_encode,A4) = aa_set_nat_nat(nat_set_encode,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_4941_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(nat_set_encode,A4) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_4942_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),X6)
           => ? [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),X6)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X5)),(Xa)) ) )
       => ~ aa_set_o_o(finite_finite_o,X6) ) ) ).

% infinite_growing
tff(fact_4943_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),X6)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real(Xa),X6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X6) ) ) ).

% infinite_growing
tff(fact_4944_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),X6)
           => ? [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),X6)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Xa) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X6) ) ) ).

% infinite_growing
tff(fact_4945_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( aa_set_num_o(member_num(X5),X6)
           => ? [Xa: num] :
                ( aa_set_num_o(member_num(Xa),X6)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X6) ) ) ).

% infinite_growing
tff(fact_4946_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),X6)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X6)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X6) ) ) ).

% infinite_growing
tff(fact_4947_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),X6)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X6)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X6) ) ) ).

% infinite_growing
tff(fact_4948_ex__min__if__finite,axiom,
    ! [S2: set_o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( S2 != bot_bot_set_o )
       => ? [X5: $o] :
            ( aa_set_o_o(member_o((X5)),S2)
            & ~ ? [Xa: $o] :
                  ( aa_set_o_o(member_o((Xa)),S2)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),(X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4949_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ? [X5: real] :
            ( aa_set_real_o(member_real(X5),S2)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X5) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4950_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S2)
     => ( ( S2 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),S2)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S2)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X5) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4951_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( aa_set_num_o(finite_finite_num,S2)
     => ( ( S2 != bot_bot_set_num )
       => ? [X5: num] :
            ( aa_set_num_o(member_num(X5),S2)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S2)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X5) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4952_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),S2)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S2)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X5) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4953_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ? [X5: int] :
            ( aa_set_int_o(member_int(X5),S2)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S2)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X5) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4954_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4955_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_4956_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_4957_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4958_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4959_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( 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))),Nb)) = 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))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4960_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4961_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4962_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4963_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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))),Nb)) = 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))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4964_option_Osize__gen_I2_J,axiom,
    ! [X3: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(X3,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X3,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4965_option_Osize__gen_I2_J,axiom,
    ! [X3: fun_nat_nat,X2: nat] : size_option_nat(X3,some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X3,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4966_option_Osize__gen_I2_J,axiom,
    ! [X3: fun_num_nat,X2: num] : size_option_num(X3,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X3,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4967_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4968_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4969_set__decode__Suc,axiom,
    ! [Nb: nat,X3: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,X3))
    <=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_4970_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(R2),C)) != aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4971_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4972_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4973_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4974_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_4975_dvd__1__left,axiom,
    ! [K2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2) ).

% dvd_1_left
tff(fact_4976_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_4977_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4978_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4979_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4980_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4981_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4982_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4983_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4984_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4985_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4986_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4987_set__decode__inverse,axiom,
    ! [Nb: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,Nb)) = Nb ).

% set_decode_inverse
tff(fact_4988_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4989_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(aa_real_real(times_times_real(C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4990_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4991_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4992_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4993_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4994_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4995_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4996_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4997_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4998_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_4999_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_5000_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_5001_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_5002_signed__take__bit__numeral__of__1,axiom,
    ! [K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K2)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_5003_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_5004_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_5005_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_5006_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_5007_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_5008_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_5009_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_5010_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_5011_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_5012_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_5013_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_5014_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_5015_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_5016_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_5017_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_5018_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_5019_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_5020_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_5021_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_5022_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_5023_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_5024_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_5025_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_5026_set__decode__0,axiom,
    ! [X3: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X3))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3) ) ).

% set_decode_0
tff(fact_5027_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5028_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5029_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5030_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5031_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5032_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5033_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_5034_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_5035_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_5036_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_5037_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_5038_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_5039_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_5040_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_5041_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_5042_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_5043_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5044_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5045_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5046_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5047_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5048_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5049_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_5050_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_5051_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_5052_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5053_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5054_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5055_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_5056_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( 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),Nb) ) ) ).

% even_power
tff(fact_5057_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Nb))
    <=> ( 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),Nb) ) ) ).

% even_power
tff(fact_5058_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_5059_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5060_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5061_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5062_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5063_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5064_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5065_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5066_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5067_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(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))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5068_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_5069_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_5070_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_5071_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_5072_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_5073_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_5074_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_5075_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_5076_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_5077_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_5078_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_5079_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C)) ) ) ).

% dvd_add
tff(fact_5080_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ) ).

% dvd_add
tff(fact_5081_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_5082_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_5083_dvd__power__same,axiom,
    ! [X3: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X3),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X3),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).

% dvd_power_same
tff(fact_5084_dvd__power__same,axiom,
    ! [X3: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_5085_dvd__power__same,axiom,
    ! [X3: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X3),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(X3),Nb)),aa_nat_real(power_power_real(Y),Nb)) ) ).

% dvd_power_same
tff(fact_5086_dvd__power__same,axiom,
    ! [X3: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X3),Nb)),aa_nat_int(power_power_int(Y),Nb)) ) ).

% dvd_power_same
tff(fact_5087_dvd__power__same,axiom,
    ! [X3: complex,Y: complex,Nb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,X3),Y)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(X3),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ) ).

% dvd_power_same
tff(fact_5088_dvd__diff__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_5089_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_fr,A))),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_fr,B)))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ).

% subset_divisors_dvd
tff(fact_5090_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_fs,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_fs,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_5091_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ft,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ft,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_5092_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fu,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fu,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_5093_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_5094_even__signed__take__bit__iff,axiom,
    ! [Mb: 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_ri631733984087533419it_int(Mb),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_5095_finite__set__decode,axiom,
    ! [Nb: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(nat_set_decode,Nb)) ).

% finite_set_decode
tff(fact_5096_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5097_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5098_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5099_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5100_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5101_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5102_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_5103_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_5104_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% div_power
tff(fact_5105_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5106_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5107_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5108_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5109_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_5110_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5111_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5112_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5113_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5114_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_5115_dvd__power__le,axiom,
    ! [X3: code_integer,Y: code_integer,Nb: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X3),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_5116_dvd__power__le,axiom,
    ! [X3: nat,Y: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_5117_dvd__power__le,axiom,
    ! [X3: real,Y: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(X3),Nb)),aa_nat_real(power_power_real(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_5118_dvd__power__le,axiom,
    ! [X3: int,Y: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X3),Nb)),aa_nat_int(power_power_int(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_5119_dvd__power__le,axiom,
    ! [X3: complex,Y: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,X3),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(X3),Nb)),aa_nat_complex(power_power_complex(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_5120_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_5121_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_5122_dvd__diffD,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb) ) ) ) ).

% dvd_diffD
tff(fact_5123_dvd__diffD1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_5124_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_5125_dbl__def,axiom,
    ! [X3: real] : aa_real_real(neg_numeral_dbl_real,X3) = aa_real_real(plus_plus_real(X3),X3) ).

% dbl_def
tff(fact_5126_dbl__def,axiom,
    ! [X3: rat] : aa_rat_rat(neg_numeral_dbl_rat,X3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),X3) ).

% dbl_def
tff(fact_5127_dbl__def,axiom,
    ! [X3: int] : aa_int_int(neg_numeral_dbl_int,X3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),X3) ).

% dbl_def
tff(fact_5128_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_5129_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_5130_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_5131_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5132_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5133_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5134_dvd__imp__le,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% dvd_imp_le
tff(fact_5135_dvd__mult__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_5136_set__decode__def,axiom,
    ! [X3: nat] : aa_nat_set_nat(nat_set_decode,X3) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fv,X3)) ).

% set_decode_def
tff(fact_5137_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5138_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_5139_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_5140_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_5141_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5142_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5143_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5144_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_5145_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_5146_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_5147_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_5148_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_5149_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_5150_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5151_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5152_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5153_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,Mb)),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_5154_dvd__power__iff,axiom,
    ! [X3: code_integer,Mb: nat,Nb: nat] :
      ( ( X3 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X3),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(X3),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X3),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_5155_dvd__power__iff,axiom,
    ! [X3: nat,Mb: nat,Nb: nat] :
      ( ( X3 != 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,X3),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_5156_dvd__power__iff,axiom,
    ! [X3: int,Mb: nat,Nb: nat] :
      ( ( X3 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X3),Mb)),aa_nat_int(power_power_int(X3),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_5157_dvd__power,axiom,
    ! [Nb: nat,X3: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X3 = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),Nb)) ) ).

% dvd_power
tff(fact_5158_dvd__power,axiom,
    ! [Nb: nat,X3: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X3 = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X3),aa_nat_rat(power_power_rat(X3),Nb)) ) ).

% dvd_power
tff(fact_5159_dvd__power,axiom,
    ! [Nb: nat,X3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X3 = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Nb)) ) ).

% dvd_power
tff(fact_5160_dvd__power,axiom,
    ! [Nb: nat,X3: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X3 = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X3),aa_nat_real(power_power_real(X3),Nb)) ) ).

% dvd_power
tff(fact_5161_dvd__power,axiom,
    ! [Nb: nat,X3: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X3 = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),aa_nat_int(power_power_int(X3),Nb)) ) ).

% dvd_power
tff(fact_5162_dvd__power,axiom,
    ! [Nb: nat,X3: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X3 = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,X3),aa_nat_complex(power_power_complex(X3),Nb)) ) ).

% dvd_power
tff(fact_5163_div2__even__ext__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( X3 = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_5164_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5165_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5166_dvd__minus__add,axiom,
    ! [Q3: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_5167_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_5168_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5169_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5170_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5171_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5172_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_5173_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_5174_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5175_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5176_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5177_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5178_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5179_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5180_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5181_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_5182_dvd__power__iff__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_5183_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_5184_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5185_even__unset__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5186_even__unset__bit__iff,axiom,
    ! [Mb: 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_se4203085406695923979it_int(Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5187_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5188_even__set__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5189_even__set__bit__iff,axiom,
    ! [Mb: 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_se7879613467334960850it_int(Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5190_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5191_even__flip__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5192_even__flip__bit__iff,axiom,
    ! [Mb: 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_se2159334234014336723it_int(Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5193_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_diff_iff
tff(fact_5194_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_5195_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_5196_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_5197_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_5198_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_5199_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_5200_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_5201_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_5202_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_5203_zero__le__even__power,axiom,
    ! [Nb: 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))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb)) ) ).

% zero_le_even_power
tff(fact_5204_zero__le__even__power,axiom,
    ! [Nb: 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))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_5205_zero__le__even__power,axiom,
    ! [Nb: 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))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_even_power
tff(fact_5206_zero__le__odd__power,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_5207_zero__le__odd__power,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_5208_zero__le__odd__power,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_5209_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_5210_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_5211_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_5212_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_5213_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5214_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( 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,A4))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_5215_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_5216_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_5217_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_5218_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_5219_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) != aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5220_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5221_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5222_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5223_crossproduct__eq,axiom,
    ! [W: real,Y: real,X3: real,Z: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Y)),aa_real_real(times_times_real(X3),Z)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Z)),aa_real_real(times_times_real(X3),Y)) )
    <=> ( ( W = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5224_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X3: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y)) )
    <=> ( ( W = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5225_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y)) )
    <=> ( ( W = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5226_crossproduct__eq,axiom,
    ! [W: int,Y: int,X3: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X3),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y)) )
    <=> ( ( W = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5227_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( 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))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_5228_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( 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))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_5229_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( 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))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_5230_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5231_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5232_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5233_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5234_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,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))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5235_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,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))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5236_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5237_option_Osize__gen_I1_J,axiom,
    ! [X3: fun_nat_nat] : size_option_nat(X3,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5238_option_Osize__gen_I1_J,axiom,
    ! [X3: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X3,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5239_option_Osize__gen_I1_J,axiom,
    ! [X3: fun_num_nat] : size_option_num(X3,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5240_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5241_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5242_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5243_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5244_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5245_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5246_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,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,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5247_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_5248_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_5249_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_5250_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_5251_inf__period_I4_J,axiom,
    ! [D: code_integer,D2: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D2)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D2)),Ta)) ) ) ).

% inf_period(4)
tff(fact_5252_inf__period_I4_J,axiom,
    ! [D: real,D2: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D2)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_5253_inf__period_I4_J,axiom,
    ! [D: rat,D2: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D2)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_5254_inf__period_I4_J,axiom,
    ! [D: int,D2: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_5255_inf__period_I3_J,axiom,
    ! [D: code_integer,D2: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D2)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D2)),Ta)) ) ) ).

% inf_period(3)
tff(fact_5256_inf__period_I3_J,axiom,
    ! [D: real,D2: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D2)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_5257_inf__period_I3_J,axiom,
    ! [D: rat,D2: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D2)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_5258_inf__period_I3_J,axiom,
    ! [D: int,D2: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_5259_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X4,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5260_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(times_times_real(L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(plus_plus_real(X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5261_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5262_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5263_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),zero_zero_int))
          & aa_int_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_5264_vebt__buildup_Oelims,axiom,
    ! [X3: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va3: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,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.elims
tff(fact_5265_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_nat_o,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_o(P,X3)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,X3)),I)) ) ) ).

% intind
tff(fact_5266_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_VEBT_VEBT_o,X3: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBT_o(P,X3)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,X3),I)) ) ) ).

% intind
tff(fact_5267_length__replicate,axiom,
    ! [Nb: nat,X3: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,X3)) = Nb ).

% length_replicate
tff(fact_5268_length__replicate,axiom,
    ! [Nb: nat,X3: $o] : size_size_list_o(replicate_o(Nb,(X3))) = Nb ).

% length_replicate
tff(fact_5269_length__replicate,axiom,
    ! [Nb: nat,X3: nat] : size_size_list_nat(replicate_nat(Nb,X3)) = Nb ).

% length_replicate
tff(fact_5270_length__replicate,axiom,
    ! [Nb: nat,X3: int] : size_size_list_int(replicate_int(Nb,X3)) = Nb ).

% length_replicate
tff(fact_5271_in__set__replicate,axiom,
    ! [X3: complex,Nb: nat,Y: complex] :
      ( aa_set_complex_o(member_complex(X3),set_complex2(replicate_complex(Nb,Y)))
    <=> ( ( X3 = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5272_in__set__replicate,axiom,
    ! [X3: real,Nb: nat,Y: real] :
      ( aa_set_real_o(member_real(X3),set_real2(replicate_real(Nb,Y)))
    <=> ( ( X3 = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5273_in__set__replicate,axiom,
    ! [X3: $o,Nb: nat,Y: $o] :
      ( aa_set_o_o(member_o((X3)),set_o2(replicate_o(Nb,(Y))))
    <=> ( ( (X3)
        <=> (Y) )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5274_in__set__replicate,axiom,
    ! [X3: int,Nb: nat,Y: int] :
      ( aa_set_int_o(member_int(X3),set_int2(replicate_int(Nb,Y)))
    <=> ( ( X3 = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5275_in__set__replicate,axiom,
    ! [X3: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(X3),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( X3 = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5276_in__set__replicate,axiom,
    ! [X3: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( X3 = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5277_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5278_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5279_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5280_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5281_set__replicate,axiom,
    ! [Nb: nat,X3: produc3843707927480180839at_nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(Nb,X3)) = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) ) ).

% set_replicate
tff(fact_5282_set__replicate,axiom,
    ! [Nb: nat,X3: vEBT_VEBT] :
      ( ( Nb != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,X3)) = insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_5283_set__replicate,axiom,
    ! [Nb: nat,X3: product_prod_nat_nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Nb,X3)) = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ) ).

% set_replicate
tff(fact_5284_set__replicate,axiom,
    ! [Nb: nat,X3: $o] :
      ( ( Nb != zero_zero_nat )
     => ( set_o2(replicate_o(Nb,(X3))) = insert_o((X3),bot_bot_set_o) ) ) ).

% set_replicate
tff(fact_5285_set__replicate,axiom,
    ! [Nb: nat,X3: nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_nat2(replicate_nat(Nb,X3)) = insert_nat(X3,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_5286_set__replicate,axiom,
    ! [Nb: nat,X3: int] :
      ( ( Nb != zero_zero_nat )
     => ( set_int2(replicate_int(Nb,X3)) = insert_int(X3,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_5287_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_5288_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
         => ( X5 = X3 ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_5289_replicate__length__same,axiom,
    ! [Xs: list_o,X3: $o] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),set_o2(Xs))
         => ( (X5)
          <=> (X3) ) )
     => ( replicate_o(size_size_list_o(Xs),(X3)) = Xs ) ) ).

% replicate_length_same
tff(fact_5290_replicate__length__same,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
         => ( X5 = X3 ) )
     => ( replicate_nat(size_size_list_nat(Xs),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_5291_replicate__length__same,axiom,
    ! [Xs: list_int,X3: int] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(Xs))
         => ( X5 = X3 ) )
     => ( replicate_int(size_size_list_int(Xs),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_5292_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,X3: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y3: complex] :
            ( aa_set_complex_o(member_complex(Y3),set_complex2(Xs))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_complex(Nb,X3) ) ) ) ).

% replicate_eqI
tff(fact_5293_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,X3: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y3: real] :
            ( aa_set_real_o(member_real(Y3),set_real2(Xs))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_real(Nb,X3) ) ) ) ).

% replicate_eqI
tff(fact_5294_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y3: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y3,set_VEBT_VEBT2(Xs))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,X3) ) ) ) ).

% replicate_eqI
tff(fact_5295_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,X3: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y3: $o] :
            ( aa_set_o_o(member_o((Y3)),set_o2(Xs))
           => ( (Y3)
            <=> (X3) ) )
       => ( Xs = replicate_o(Nb,(X3)) ) ) ) ).

% replicate_eqI
tff(fact_5296_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,X3: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),set_nat2(Xs))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_nat(Nb,X3) ) ) ) ).

% replicate_eqI
tff(fact_5297_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,X3: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),set_int2(Xs))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_int(Nb,X3) ) ) ) ).

% replicate_eqI
tff(fact_5298_set__replicate__Suc,axiom,
    ! [Nb: nat,X3: produc3843707927480180839at_nat] : set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(aa_nat_nat(suc,Nb),X3)) = insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ).

% set_replicate_Suc
tff(fact_5299_set__replicate__Suc,axiom,
    ! [Nb: nat,X3: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),X3)) = insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_5300_set__replicate__Suc,axiom,
    ! [Nb: nat,X3: product_prod_nat_nat] : set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(aa_nat_nat(suc,Nb),X3)) = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ).

% set_replicate_Suc
tff(fact_5301_set__replicate__Suc,axiom,
    ! [Nb: nat,X3: $o] : set_o2(replicate_o(aa_nat_nat(suc,Nb),(X3))) = insert_o((X3),bot_bot_set_o) ).

% set_replicate_Suc
tff(fact_5302_set__replicate__Suc,axiom,
    ! [Nb: nat,X3: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),X3)) = insert_nat(X3,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_5303_set__replicate__Suc,axiom,
    ! [Nb: nat,X3: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),X3)) = insert_int(X3,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_5304_set__replicate__conv__if,axiom,
    ! [Nb: nat,X3: produc3843707927480180839at_nat] :
      set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(Nb,X3)) = $ite(Nb = zero_zero_nat,bot_bo228742789529271731at_nat,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)) ).

% set_replicate_conv_if
tff(fact_5305_set__replicate__conv__if,axiom,
    ! [Nb: nat,X3: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,X3)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_5306_set__replicate__conv__if,axiom,
    ! [Nb: nat,X3: product_prod_nat_nat] :
      set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Nb,X3)) = $ite(Nb = zero_zero_nat,bot_bo2099793752762293965at_nat,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)) ).

% set_replicate_conv_if
tff(fact_5307_set__replicate__conv__if,axiom,
    ! [Nb: nat,X3: $o] :
      set_o2(replicate_o(Nb,(X3))) = $ite(Nb = zero_zero_nat,bot_bot_set_o,insert_o((X3),bot_bot_set_o)) ).

% set_replicate_conv_if
tff(fact_5308_set__replicate__conv__if,axiom,
    ! [Nb: nat,X3: nat] :
      set_nat2(replicate_nat(Nb,X3)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat(X3,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_5309_set__replicate__conv__if,axiom,
    ! [Nb: nat,X3: int] :
      set_int2(replicate_int(Nb,X3)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int(X3,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_5310_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_5311_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% minf(8)
tff(fact_5312_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% minf(8)
tff(fact_5313_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_5314_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_5315_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_5316_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% minf(6)
tff(fact_5317_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% minf(6)
tff(fact_5318_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_5319_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_5320_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_5321_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% pinf(8)
tff(fact_5322_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% pinf(8)
tff(fact_5323_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_5324_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_5325_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_5326_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% pinf(6)
tff(fact_5327_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% pinf(6)
tff(fact_5328_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_5329_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_5330_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B3),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_5331_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X5: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_5332_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X5: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_5333_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5334_minf_I10_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% minf(10)
tff(fact_5335_minf_I10_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb)) ) ) ).

% minf(10)
tff(fact_5336_minf_I10_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb)) ) ) ).

% minf(10)
tff(fact_5337_minf_I10_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% minf(10)
tff(fact_5338_minf_I10_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb)) ) ) ).

% minf(10)
tff(fact_5339_minf_I9_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% minf(9)
tff(fact_5340_minf_I9_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb)) ) ) ).

% minf(9)
tff(fact_5341_minf_I9_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb)) ) ) ).

% minf(9)
tff(fact_5342_minf_I9_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% minf(9)
tff(fact_5343_minf_I9_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb)) ) ) ).

% minf(9)
tff(fact_5344_pinf_I10_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% pinf(10)
tff(fact_5345_pinf_I10_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb)) ) ) ).

% pinf(10)
tff(fact_5346_pinf_I10_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb)) ) ) ).

% pinf(10)
tff(fact_5347_pinf_I10_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% pinf(10)
tff(fact_5348_pinf_I10_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb)) ) ) ).

% pinf(10)
tff(fact_5349_pinf_I9_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% pinf(9)
tff(fact_5350_pinf_I9_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Sb)) ) ) ).

% pinf(9)
tff(fact_5351_pinf_I9_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Sb)) ) ) ).

% pinf(9)
tff(fact_5352_pinf_I9_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% pinf(9)
tff(fact_5353_pinf_I9_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Sb)) ) ) ).

% pinf(9)
tff(fact_5354_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X5: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_5355_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5356_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5357_vebt__buildup_Opelims,axiom,
    ! [X3: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X3) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),X3)
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va3: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,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))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va3))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5358_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_5359_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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_5360_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5361_artanh__def,axiom,
    ! [X3: real] : aa_real_real(artanh_real,X3) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(one_one_real),X3)),aa_real_real(minus_minus_real(one_one_real),X3)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5362_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fw(L)),Qr) ).

% divmod_step_def
tff(fact_5363_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_fx(L)),Qr) ).

% divmod_step_def
tff(fact_5364_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_fy(L),Qr) ).

% divmod_step_def
tff(fact_5365_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_5366_take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Nb,A) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_5367_take__bit__rec,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_5368_take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = 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,Nb),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_5369_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_5370_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_5371_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_5372_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_5373_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_5374_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5375_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5376_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5377_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5378_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5379_Compl__subset__Compl__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(A4)),uminus5710092332889474511et_nat(B4))
    <=> aa_set_nat_o(ord_less_eq_set_nat(B4),A4) ) ).

% Compl_subset_Compl_iff
tff(fact_5380_Compl__anti__mono,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(B4)),uminus5710092332889474511et_nat(A4)) ) ).

% Compl_anti_mono
tff(fact_5381_compl__le__compl__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X3)),uminus5710092332889474511et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Y),X3) ) ).

% compl_le_compl_iff
tff(fact_5382_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_5383_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_5384_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_5385_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_5386_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_5387_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_5388_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_5389_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_5390_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_5391_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5392_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_5393_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_5394_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_5395_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_5396_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_5397_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_5398_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_5399_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_5400_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_5401_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_5402_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_5403_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_5404_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_5405_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_5406_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_5407_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_5408_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_5409_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_5410_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_5411_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_5412_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_5413_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5414_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5415_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5416_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5417_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5418_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_5419_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_5420_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_5421_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_5422_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_5423_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_5424_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(plus_plus_real(A),B)) = B ).

% minus_add_cancel
tff(fact_5425_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_5426_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_5427_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_5428_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_5429_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_5430_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_5431_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_5432_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_5433_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_5434_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ).

% minus_diff_eq
tff(fact_5435_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_5436_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_5437_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% minus_diff_eq
tff(fact_5438_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_5439_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_5440_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_5441_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa_num_num(bit1,Mb) != one ).

% semiring_norm(86)
tff(fact_5442_Compl__disjoint2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A4)),A4) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint2
tff(fact_5443_Compl__disjoint2,axiom,
    ! [A4: set_o] : inf_inf_set_o(uminus_uminus_set_o(A4),A4) = bot_bot_set_o ).

% Compl_disjoint2
tff(fact_5444_Compl__disjoint2,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A4)),A4) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_5445_Compl__disjoint2,axiom,
    ! [A4: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(A4),A4) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_5446_Compl__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),uminus6524753893492686040at_nat(A4)) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint
tff(fact_5447_Compl__disjoint,axiom,
    ! [A4: set_o] : inf_inf_set_o(A4,uminus_uminus_set_o(A4)) = bot_bot_set_o ).

% Compl_disjoint
tff(fact_5448_Compl__disjoint,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(A4)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_5449_Compl__disjoint,axiom,
    ! [A4: set_int] : inf_inf_set_int(A4,uminus1532241313380277803et_int(A4)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_5450_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_5451_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_5452_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_5453_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),aa_o_Code_integer(zero_n356916108424825756nteger,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5454_Diff__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),uminus6524753893492686040at_nat(B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ).

% Diff_Compl
tff(fact_5455_Diff__Compl,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),uminus5710092332889474511et_nat(B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ).

% Diff_Compl
tff(fact_5456_Compl__Diff__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A4)),B4) ).

% Compl_Diff_eq
tff(fact_5457_Compl__Diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A4)),B4) ).

% Compl_Diff_eq
tff(fact_5458_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prod_conv
tff(fact_5459_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) ).

% case_prod_conv
tff(fact_5460_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_5461_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_5462_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_5463_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_5464_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_5465_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_5466_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_5467_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_5468_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_5469_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_5470_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_5471_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_5472_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_5473_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_5474_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_5475_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_5476_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_5477_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_5478_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_5479_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_5480_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_5481_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_5482_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_5483_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_5484_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_5485_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_5486_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_5487_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_5488_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_5489_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_5490_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_5491_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_5492_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_5493_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_5494_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_5495_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_5496_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_5497_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_5498_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_5499_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_5500_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_5501_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_5502_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_5503_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_5504_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_5505_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_5506_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_5507_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_5508_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_5509_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_5510_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5511_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5512_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5513_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5514_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5515_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_5516_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(plus_plus_real(A),B) ).

% diff_minus_eq_add
tff(fact_5517_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_5518_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_5519_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_5520_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_5521_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ).

% uminus_add_conv_diff
tff(fact_5522_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_5523_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_5524_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% uminus_add_conv_diff
tff(fact_5525_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_5526_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),uminus6524753893492686040at_nat(X3)) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_5527_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_o] : inf_inf_set_o(X3,uminus_uminus_set_o(X3)) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_right
tff(fact_5528_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),uminus5710092332889474511et_nat(X3)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_5529_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_int] : inf_inf_set_int(X3,uminus1532241313380277803et_int(X3)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_5530_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X3)),X3) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_5531_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_o] : inf_inf_set_o(uminus_uminus_set_o(X3),X3) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_left
tff(fact_5532_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X3)),X3) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_5533_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(X3),X3) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_5534_inf__compl__bot__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),uminus6524753893492686040at_nat(X3))) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_right
tff(fact_5535_inf__compl__bot__right,axiom,
    ! [X3: set_o,Y: set_o] : inf_inf_set_o(X3,inf_inf_set_o(Y,uminus_uminus_set_o(X3))) = bot_bot_set_o ).

% inf_compl_bot_right
tff(fact_5536_inf__compl__bot__right,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),uminus5710092332889474511et_nat(X3))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_5537_inf__compl__bot__right,axiom,
    ! [X3: set_int,Y: set_int] : inf_inf_set_int(X3,inf_inf_set_int(Y,uminus1532241313380277803et_int(X3))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_5538_inf__compl__bot__left2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X3)),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left2
tff(fact_5539_inf__compl__bot__left2,axiom,
    ! [X3: set_o,Y: set_o] : inf_inf_set_o(X3,inf_inf_set_o(uminus_uminus_set_o(X3),Y)) = bot_bot_set_o ).

% inf_compl_bot_left2
tff(fact_5540_inf__compl__bot__left2,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X3)),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_5541_inf__compl__bot__left2,axiom,
    ! [X3: set_int,Y: set_int] : inf_inf_set_int(X3,inf_inf_set_int(uminus1532241313380277803et_int(X3),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_5542_inf__compl__bot__left1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left1
tff(fact_5543_inf__compl__bot__left1,axiom,
    ! [X3: set_o,Y: set_o] : inf_inf_set_o(uminus_uminus_set_o(X3),inf_inf_set_o(X3,Y)) = bot_bot_set_o ).

% inf_compl_bot_left1
tff(fact_5544_inf__compl__bot__left1,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_5545_inf__compl__bot__left1,axiom,
    ! [X3: set_int,Y: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(X3),inf_inf_set_int(X3,Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_5546_subset__Compl__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),uminus935396558254630718at_nat(insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)))
    <=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_5547_subset__Compl__singleton,axiom,
    ! [A4: set_complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),uminus8566677241136511917omplex(insert_complex(B,bot_bot_set_complex)))
    <=> ~ aa_set_complex_o(member_complex(B),A4) ) ).

% subset_Compl_singleton
tff(fact_5548_subset__Compl__singleton,axiom,
    ! [A4: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B),A4) ) ).

% subset_Compl_singleton
tff(fact_5549_subset__Compl__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)))
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_5550_subset__Compl__singleton,axiom,
    ! [A4: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(insert_o((B),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B)),A4) ) ).

% subset_Compl_singleton
tff(fact_5551_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A4) ) ).

% subset_Compl_singleton
tff(fact_5552_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_5553_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_5554_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_5555_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_5556_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_5557_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_5558_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(7)
tff(fact_5559_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(9)
tff(fact_5560_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_5561_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_5562_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_5563_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_5564_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_5565_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_5566_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_simps(1)
tff(fact_5567_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_simps(1)
tff(fact_5568_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2))) ).

% dbl_simps(1)
tff(fact_5569_dbl__simps_I1_J,axiom,
    ! [K2: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K2))) ).

% dbl_simps(1)
tff(fact_5570_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_simps(1)
tff(fact_5571_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_5572_add__neg__numeral__special_I7_J,axiom,
    aa_real_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_5573_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_5574_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_5575_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_5576_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_5577_add__neg__numeral__special_I8_J,axiom,
    aa_real_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_5578_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_5579_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_5580_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_5581_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5582_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5583_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5584_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5585_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5586_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5587_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5588_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5589_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5590_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5591_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_5592_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_5593_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_5594_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_5595_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_5596_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5597_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5598_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5599_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5600_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5601_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = $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,V2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2)),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(2)
tff(fact_5602_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2)),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(2)
tff(fact_5603_max__number__of_I2_J,axiom,
    ! [U: num,V2: 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,V2))) = $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,V2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_5604_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_5605_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V2)) = $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,V2)),aa_num_real(numeral_numeral_real,V2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(3)
tff(fact_5606_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V2)),aa_num_Code_integer(numera6620942414471956472nteger,V2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(3)
tff(fact_5607_max__number__of_I3_J,axiom,
    ! [U: num,V2: 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,V2)) = $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,V2)),aa_num_rat(numeral_numeral_rat,V2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_5608_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,V2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_5609_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      aa_real_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,V2))) = $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,V2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(4)
tff(fact_5610_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(4)
tff(fact_5611_max__number__of_I4_J,axiom,
    ! [U: num,V2: 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,V2))) = $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,V2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_5612_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_5613_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W)))),Y) ).

% semiring_norm(168)
tff(fact_5614_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W)))),Y) ).

% semiring_norm(168)
tff(fact_5615_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V2))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W)))),Y) ).

% semiring_norm(168)
tff(fact_5616_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W))),Y) ).

% semiring_norm(168)
tff(fact_5617_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W)))),Y) ).

% semiring_norm(168)
tff(fact_5618_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5619_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5620_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5621_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5622_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5623_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5624_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5625_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5626_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5627_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5628_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_5629_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_5630_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% semiring_norm(4)
tff(fact_5631_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_5632_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one)) ).

% semiring_norm(8)
tff(fact_5633_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_5634_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(170)
tff(fact_5635_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Y)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(170)
tff(fact_5636_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(170)
tff(fact_5637_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(170)
tff(fact_5638_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(170)
tff(fact_5639_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(171)
tff(fact_5640_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V2)),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(171)
tff(fact_5641_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(171)
tff(fact_5642_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(171)
tff(fact_5643_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W)))),Y) ).

% semiring_norm(171)
tff(fact_5644_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Y) ).

% semiring_norm(172)
tff(fact_5645_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Y) ).

% semiring_norm(172)
tff(fact_5646_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Y) ).

% semiring_norm(172)
tff(fact_5647_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Y) ).

% semiring_norm(172)
tff(fact_5648_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W))),Y) ).

% semiring_norm(172)
tff(fact_5649_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5650_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5651_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5652_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5653_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5654_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5655_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5656_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5657_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5658_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5659_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5660_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5661_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5662_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5663_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5664_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5665_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5666_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5667_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5668_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5669_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5670_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5671_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5672_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_5673_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_5674_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_5675_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: 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,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5676_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5677_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: 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,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5678_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5679_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5680_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5681_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5682_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5683_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5684_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5685_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5686_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5687_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5688_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5689_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5690_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5691_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5692_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5693_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5694_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5695_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5696_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5697_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5698_power2__minus,axiom,
    ! [A: real] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5699_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5700_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5701_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5702_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5703_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5704_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5705_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5706_add__neg__numeral__special_I9_J,axiom,
    aa_real_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_5707_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_5708_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5709_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_5710_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5711_diff__numeral__special_I10_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5712_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_5713_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5714_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_5715_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5716_diff__numeral__special_I11_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5717_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_5718_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5719_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_5720_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_5721_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_5722_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_5723_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5724_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_5725_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5726_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5727_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5728_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5729_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5730_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5731_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_5732_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_5733_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_5734_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5735_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(power_power_real(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5736_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5737_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5738_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5739_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5740_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5741_power__minus__odd,axiom,
    ! [Nb: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5742_power__minus__odd,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5743_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5744_even__take__bit__eq,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Nb,A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5745_even__take__bit__eq,axiom,
    ! [Nb: 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,Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5746_even__take__bit__eq,axiom,
    ! [Nb: 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(Nb),A))
    <=> ( ( Nb = 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_5747_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5748_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_5749_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5750_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5751_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5752_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5753_diff__numeral__special_I3_J,axiom,
    ! [Nb: 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,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5754_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5755_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5756_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5757_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5758_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5759_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5760_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_5761_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5762_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5763_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_5764_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_5765_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5766_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_5767_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_5768_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_5769_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_5770_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_5771_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_5772_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_5773_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_5774_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_5775_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_5776_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_5777_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_5778_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_5779_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_5780_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_5781_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_5782_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5783_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_5784_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_5785_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5786_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(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_5787_one__div__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5788_one__div__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5789_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_o_Code_integer(zero_n356916108424825756nteger,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5790_bits__1__div__exp,axiom,
    ! [Nb: 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))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5791_bits__1__div__exp,axiom,
    ! [Nb: 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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5792_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_o_Code_integer(zero_n356916108424825756nteger,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5793_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : bit_se1745604003318907178nteger(Mb,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_5794_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_5795_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_5796_take__bit__of__2,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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))),Nb))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5797_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_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))),Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5798_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5799_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_5800_one__mod__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5801_one__mod__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5802_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5803_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_5804_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_5805_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_5806_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_5807_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_5808_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_5809_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_5810_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_5811_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_5812_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_5813_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_5814_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_fz(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5815_prod_Ocase__distrib,axiom,
    ! [H: fun_o_nat,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] : aa_o_nat(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun428955956450291160at_nat(aTP_Lamm_ga(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5816_prod_Ocase__distrib,axiom,
    ! [H: fun_nat_o,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] :
      ( aa_nat_o(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun5171641919285310978_nat_o(aTP_Lamm_gb(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5817_prod_Ocase__distrib,axiom,
    ! [H: fun_nat_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_gc(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5818_prod_Ocase__distrib,axiom,
    ! [H: fun_o_3115027942823555909_nat_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] : aa_o_f8445886544701262710_nat_o(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun5069079742692966690_nat_o(aTP_Lamm_gd(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5819_prod_Ocase__distrib,axiom,
    ! [H: fun_na3970208099681795815_nat_o,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat7695548938202568488_nat_o(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun2100511394839469148_nat_o(aTP_Lamm_ge(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5820_prod_Ocase__distrib,axiom,
    ! [H: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun7237989844890293688_nat_o(aTP_Lamm_gf(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5821_prod_Ocase__distrib,axiom,
    ! [H: fun_fu1115759055147092071_o_nat,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun1432809138467554728_o_nat(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun3680403850923830236at_nat(aTP_Lamm_gg(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5822_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_gh(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5823_prod_Ocase__distrib,axiom,
    ! [H: fun_o_4034989691794805934at_nat,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] : aa_o_f1925640989104809311at_nat(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod)) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aa_fun8042081135417832459at_nat(aTP_Lamm_gi(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5824_power__minus__Bit1,axiom,
    ! [X3: int,K2: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5825_power__minus__Bit1,axiom,
    ! [X3: real,K2: num] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5826_power__minus__Bit1,axiom,
    ! [X3: complex,K2: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5827_power__minus__Bit1,axiom,
    ! [X3: code_integer,K2: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5828_power__minus__Bit1,axiom,
    ! [X3: rat,K2: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5829_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_5830_take__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_5831_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_5832_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B: int,Mb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(Mb),A) = aa_int_int(bit_se2923211474154528505it_int(Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_5833_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5834_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_5835_compl__le__swap2,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X3)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X3)),Y) ) ).

% compl_le_swap2
tff(fact_5836_compl__le__swap1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X3))
     => aa_set_nat_o(ord_less_eq_set_nat(X3),uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_5837_compl__mono,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X3),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X3)) ) ).

% compl_mono
tff(fact_5838_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_5839_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_5840_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_5841_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_5842_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_5843_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_5844_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_5845_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_5846_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_5847_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_5848_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_5849_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_5850_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_5851_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_5852_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_5853_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_5854_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_5855_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_5856_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_5857_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_5858_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_5859_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_5860_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5861_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_real(numeral_numeral_real,Mb) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5862_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5863_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_Code_integer(numera6620942414471956472nteger,Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5864_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Mb) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5865_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5866_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5867_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5868_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)) != aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5869_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5870_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_5871_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_5872_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_5873_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_5874_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_5875_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_5876_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_5877_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_5878_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_5879_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_5880_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_5881_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(uminus_uminus_real,A4) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K2)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_5882_group__cancel_Oneg1,axiom,
    ! [A4: complex,K2: complex,A: complex] :
      ( ( A4 = aa_complex_complex(plus_plus_complex(K2),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_5883_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4 = plus_p5714425477246183910nteger(K2,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_5884_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_5885_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_5886_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_5887_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_5888_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_5889_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_5890_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_o,X1: nat,X2: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,X1),X2) ) ).

% old.prod.case
tff(fact_5891_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_nat,X1: nat,X2: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5892_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5893_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).

% old.prod.case
tff(fact_5894_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).

% old.prod.case
tff(fact_5895_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_nat_fun_nat_o,Z: product_prod_nat_nat] :
      ( aa_o_o(Q,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(P),Z))
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) )
           => ~ aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(P,X5),Y3)) ) ) ).

% case_prodE2
tff(fact_5896_case__prodE2,axiom,
    ! [Q: fun_nat_o,P: fun_nat_fun_nat_nat,Z: product_prod_nat_nat] :
      ( aa_nat_o(Q,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(P),Z))
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) )
           => ~ aa_nat_o(Q,aa_nat_nat(aa_nat_fun_nat_nat(P,X5),Y3)) ) ) ).

% case_prodE2
tff(fact_5897_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z))
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X5),Y3)) ) ) ).

% case_prodE2
tff(fact_5898_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z))
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X5),Y3)) ) ) ).

% case_prodE2
tff(fact_5899_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X5: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X5),Y3)) ) ) ).

% case_prodE2
tff(fact_5900_case__prod__eta,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(aTP_Lamm_gj(F)) = F ).

% case_prod_eta
tff(fact_5901_case__prod__eta,axiom,
    ! [F: fun_Pr2602117396471556196at_nat] : produc6842872674320459806at_nat(aTP_Lamm_gk(F)) = F ).

% case_prod_eta
tff(fact_5902_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_gl(F)) = F ).

% case_prod_eta
tff(fact_5903_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_gm(F)) = F ).

% case_prod_eta
tff(fact_5904_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_gn(F)) = F ).

% case_prod_eta
tff(fact_5905_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_o,G: fun_Pr4796943543331071514_nat_o] :
      ( ! [X5: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(F,X5),Y3)
        <=> aa_Pro6150637604445964947_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3)) )
     => ( produc6081775807080527818_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5906_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_nat,G: fun_Pr2602117396471556196at_nat] :
      ( ! [X5: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3) = aa_Pro4243173770044397973at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3))
     => ( produc6842872674320459806at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5907_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X5: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X5),Y3) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5908_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X5: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X5),Y3) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5909_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X5: int,Y3: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X5),Y3) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Y3))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5910_Collect__imp__eq,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] : collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_go(P),Q)) = sup_sup_set_complex(uminus8566677241136511917omplex(collect_complex(P)),collect_complex(Q)) ).

% Collect_imp_eq
tff(fact_5911_Collect__imp__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] : collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_gp(P),Q)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(collec3392354462482085612at_nat(P))),collec3392354462482085612at_nat(Q)) ).

% Collect_imp_eq
tff(fact_5912_Collect__imp__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_gq(P),Q)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(P)),collect_set_nat(Q)) ).

% Collect_imp_eq
tff(fact_5913_Collect__imp__eq,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_gr(P),Q)) = sup_sup_set_list_nat(uminus3195874150345416415st_nat(collect_list_nat(P)),collect_list_nat(Q)) ).

% Collect_imp_eq
tff(fact_5914_Collect__imp__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_gs(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(collect_nat(P))),collect_nat(Q)) ).

% Collect_imp_eq
tff(fact_5915_Collect__imp__eq,axiom,
    ! [P: fun_Pr3059359313727237556_nat_o,Q: fun_Pr3059359313727237556_nat_o] : collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_gt(P),Q)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(collec6321179662152712658at_nat(P))),collec6321179662152712658at_nat(Q)) ).

% Collect_imp_eq
tff(fact_5916_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_5917_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_5918_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_5919_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_5920_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(P))) ).

% zero_less_eq_of_bool
tff(fact_5921_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_5922_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_5923_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_5924_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_5925_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_5926_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5927_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5928_signed__take__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Mb)),A) ).

% signed_take_bit_take_bit
tff(fact_5929_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5930_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5931_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5932_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5933_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5934_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5935_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5936_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5937_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5938_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5939_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5940_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5941_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5942_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5943_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5944_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5945_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5946_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5947_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5948_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5949_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5950_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] : Y != aa_num_num(bit0,X22)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_5951_xor__num_Ocases,axiom,
    ! [X3: product_prod_num_num] :
      ( ( X3 != product_Pair_num_num(one,one) )
     => ( ! [N: num] : X3 != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : X3 != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M: num] : X3 != product_Pair_num_num(aa_num_num(bit0,M),one)
           => ( ! [M: num,N: num] : X3 != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))
             => ( ! [M: num,N: num] : X3 != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))
               => ( ! [M: num] : X3 != product_Pair_num_num(aa_num_num(bit1,M),one)
                 => ( ! [M: num,N: num] : X3 != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))
                   => ~ ! [M: num,N: num] : X3 != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5952_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_5953_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_5954_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_5955_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_5956_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_5957_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_5958_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_5959_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_5960_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_5961_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5962_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5963_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5964_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5965_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_5966_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5967_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_5968_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_5969_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_5970_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_5971_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_5972_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_5973_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_5974_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5975_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5976_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5977_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5978_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5979_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5980_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5981_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5982_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5983_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5984_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5985_numeral__times__minus__swap,axiom,
    ! [W: num,X3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X3)) = aa_int_int(aa_int_fun_int_int(times_times_int,X3),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5986_numeral__times__minus__swap,axiom,
    ! [W: num,X3: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,X3)) = aa_real_real(times_times_real(X3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_5987_numeral__times__minus__swap,axiom,
    ! [W: num,X3: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,X3)) = aa_complex_complex(times_times_complex(X3),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5988_numeral__times__minus__swap,axiom,
    ! [W: num,X3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X3),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))) ).

% numeral_times_minus_swap
tff(fact_5989_numeral__times__minus__swap,axiom,
    ! [W: num,X3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,X3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_5990_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5991_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5992_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5993_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_Code_integer(numera6620942414471956472nteger,Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5994_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5995_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_5996_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_5997_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_5998_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% one_neq_neg_numeral
tff(fact_5999_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_6000_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6001_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6002_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6003_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6004_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6005_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_6006_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_6007_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_6008_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_6009_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_6010_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6011_group__cancel_Osub2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(plus_plus_real(K2),B) )
     => ( aa_real_real(minus_minus_real(A),B4) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K2)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6012_group__cancel_Osub2,axiom,
    ! [B4: complex,K2: complex,B: complex,A: complex] :
      ( ( B4 = aa_complex_complex(plus_plus_complex(K2),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6013_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B4 = plus_p5714425477246183910nteger(K2,B) )
     => ( minus_8373710615458151222nteger(A,B4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_6014_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6015_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_6016_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se4203085406695923979it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se4203085406695923979it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_6017_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_6018_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se7879613467334960850it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se7879613467334960850it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_6019_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_6020_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_6021_inf__cancel__left2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: 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,uminus6524753893492686040at_nat(X3)),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),B)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left2
tff(fact_6022_inf__cancel__left2,axiom,
    ! [X3: set_o,A: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(uminus_uminus_set_o(X3),A),inf_inf_set_o(X3,B)) = bot_bot_set_o ).

% inf_cancel_left2
tff(fact_6023_inf__cancel__left2,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X3)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),B)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_6024_inf__cancel__left2,axiom,
    ! [X3: set_int,A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(uminus1532241313380277803et_int(X3),A),inf_inf_set_int(X3,B)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_6025_inf__cancel__left1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: 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,X3),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X3)),B)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left1
tff(fact_6026_inf__cancel__left1,axiom,
    ! [X3: set_o,A: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(X3,A),inf_inf_set_o(uminus_uminus_set_o(X3),B)) = bot_bot_set_o ).

% inf_cancel_left1
tff(fact_6027_inf__cancel__left1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X3)),B)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_6028_inf__cancel__left1,axiom,
    ! [X3: set_int,A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(X3,A),inf_inf_set_int(uminus1532241313380277803et_int(X3),B)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_6029_subset__Compl__self__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(A4))
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6030_subset__Compl__self__eq,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(A4))
    <=> ( A4 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_6031_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(A4))
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_6032_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),uminus5710092332889474511et_nat(A4))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6033_Compl__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(A4)),uminus6524753893492686040at_nat(B4)) ).

% Compl_Int
tff(fact_6034_Compl__Int,axiom,
    ! [A4: set_nat,B4: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A4)),uminus5710092332889474511et_nat(B4)) ).

% Compl_Int
tff(fact_6035_Compl__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A4)),uminus935396558254630718at_nat(B4)) ).

% Compl_Int
tff(fact_6036_Compl__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A4)),uminus6524753893492686040at_nat(B4)) ).

% Compl_Un
tff(fact_6037_Compl__Un,axiom,
    ! [A4: set_nat,B4: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A4)),uminus5710092332889474511et_nat(B4)) ).

% Compl_Un
tff(fact_6038_Compl__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(A4)),uminus935396558254630718at_nat(B4)) ).

% Compl_Un
tff(fact_6039_Diff__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),uminus6524753893492686040at_nat(B4)) ).

% Diff_eq
tff(fact_6040_Diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(B4)) ).

% Diff_eq
tff(fact_6041_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_6042_take__bit__signed__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Mb),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_6043_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6044_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6045_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6046_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6047_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_6048_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_6049_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_6050_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_6051_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_6052_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_6053_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_6054_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_6055_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6056_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6057_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6058_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6059_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_6060_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_6061_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_6062_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_6063_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_6064_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_6065_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_6066_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_6067_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_6068_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_6069_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = 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,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_6070_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_6071_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = 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,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_6072_not__one__le__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6073_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6074_not__one__le__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6075_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6076_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_6077_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_6078_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_6079_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_6080_neg__numeral__le__neg__one,axiom,
    ! [Mb: 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,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_6081_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_6082_neg__numeral__le__neg__one,axiom,
    ! [Mb: 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,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_6083_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_6084_neg__one__le__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_le_numeral
tff(fact_6085_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Mb)) ).

% neg_one_le_numeral
tff(fact_6086_neg__one__le__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_le_numeral
tff(fact_6087_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_le_numeral
tff(fact_6088_neg__numeral__le__one,axiom,
    ! [Mb: 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,Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_6089_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_6090_neg__numeral__le__one,axiom,
    ! [Mb: 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,Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_6091_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_6092_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_6093_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_6094_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_6095_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_6096_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_less_numeral
tff(fact_6097_neg__one__less__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_less_numeral
tff(fact_6098_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Mb)) ).

% neg_one_less_numeral
tff(fact_6099_neg__one__less__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_less_numeral
tff(fact_6100_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_6101_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_6102_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_6103_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_6104_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6105_not__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6106_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6107_not__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6108_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6109_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6110_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6111_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6112_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_6113_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_6114_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_6115_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_6116_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_6117_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_6118_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_6119_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_6120_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_6121_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_6122_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_6123_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_6124_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6125_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6126_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_6127_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_6128_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_minus
tff(fact_6129_power__minus,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(power_power_real(A),Nb)) ).

% power_minus
tff(fact_6130_power__minus,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_minus
tff(fact_6131_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_6132_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_6133_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6134_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6135_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6136_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6137_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6138_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6139_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6140_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6141_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6142_inf__shunt,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),uminus6524753893492686040at_nat(Y)) ) ).

% inf_shunt
tff(fact_6143_inf__shunt,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( inf_inf_set_o(X3,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(X3),uminus_uminus_set_o(Y)) ) ).

% inf_shunt
tff(fact_6144_inf__shunt,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( inf_inf_set_int(X3,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(X3),uminus1532241313380277803et_int(Y)) ) ).

% inf_shunt
tff(fact_6145_inf__shunt,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(X3),uminus5710092332889474511et_nat(Y)) ) ).

% inf_shunt
tff(fact_6146_power__minus__Bit0,axiom,
    ! [X3: int,K2: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_6147_power__minus__Bit0,axiom,
    ! [X3: real,K2: num] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_6148_power__minus__Bit0,axiom,
    ! [X3: complex,K2: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_6149_power__minus__Bit0,axiom,
    ! [X3: code_integer,K2: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_6150_power__minus__Bit0,axiom,
    ! [X3: rat,K2: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_6151_sup__neg__inf,axiom,
    ! [P2: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,P2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q3),R2))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P2),uminus6524753893492686040at_nat(Q3))),R2) ) ).

% sup_neg_inf
tff(fact_6152_sup__neg__inf,axiom,
    ! [P2: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(P2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Q3),R2))
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,P2),uminus935396558254630718at_nat(Q3))),R2) ) ).

% sup_neg_inf
tff(fact_6153_sup__neg__inf,axiom,
    ! [P2: set_nat,Q3: set_nat,R2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(P2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q3),R2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P2),uminus5710092332889474511et_nat(Q3))),R2) ) ).

% sup_neg_inf
tff(fact_6154_shunt2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),uminus6524753893492686040at_nat(Y))),Z)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) ) ).

% shunt2
tff(fact_6155_shunt2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),uminus935396558254630718at_nat(Y))),Z)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).

% shunt2
tff(fact_6156_shunt2,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),uminus5710092332889474511et_nat(Y))),Z)
    <=> aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).

% shunt2
tff(fact_6157_shunt1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y)),Z)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_6158_shunt1,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),Y)),Z)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_6159_shunt1,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y)),Z)
    <=> aa_set_nat_o(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Y)),Z)) ) ).

% shunt1
tff(fact_6160_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = 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,Nb),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_6161_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = 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(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_6162_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6163_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6164_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A4,B4) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6165_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),uminus5710092332889474511et_nat(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6166_Compl__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(insert9069300056098147895at_nat(X3,A4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,uminus935396558254630718at_nat(A4)),insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)) ).

% Compl_insert
tff(fact_6167_Compl__insert,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X3,A4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,uminus6524753893492686040at_nat(A4)),insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)) ).

% Compl_insert
tff(fact_6168_Compl__insert,axiom,
    ! [X3: $o,A4: set_o] : uminus_uminus_set_o(insert_o((X3),A4)) = minus_minus_set_o(uminus_uminus_set_o(A4),insert_o((X3),bot_bot_set_o)) ).

% Compl_insert
tff(fact_6169_Compl__insert,axiom,
    ! [X3: int,A4: set_int] : uminus1532241313380277803et_int(insert_int(X3,A4)) = minus_minus_set_int(uminus1532241313380277803et_int(A4),insert_int(X3,bot_bot_set_int)) ).

% Compl_insert
tff(fact_6170_Compl__insert,axiom,
    ! [X3: nat,A4: set_nat] : uminus5710092332889474511et_nat(insert_nat(X3,A4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,uminus5710092332889474511et_nat(A4)),insert_nat(X3,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_6171_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_6172_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_6173_take__bit__numeral__minus__1__eq,axiom,
    ! [K2: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K2)),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_6174_take__bit__numeral__minus__1__eq,axiom,
    ! [K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,K2)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K2))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_6175_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_6176_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_6177_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_6178_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_6179_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_6180_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6181_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(aa_real_real(times_times_real(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6182_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6183_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6184_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6185_take__bit__minus__small__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K2)) = 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))),Nb)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_6186_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_6187_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_6188_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6189_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6190_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6191_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6192_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6193_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6194_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6195_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6196_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6197_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6198_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6199_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6200_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6201_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6202_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6203_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X3),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,X3)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6204_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X3),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X3)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6205_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X3),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6206_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_6207_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(times_times_real(aa_real_real(times_times_real(A),A)),A) ).

% power3_eq_cube
tff(fact_6208_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ).

% power3_eq_cube
tff(fact_6209_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_6210_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_6211_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_6212_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_6213_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_6214_power2__eq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6215_power2__eq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6216_power2__eq__iff,axiom,
    ! [X3: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6217_power2__eq__iff,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6218_power2__eq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6219_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_6220_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_6221_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_6222_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(Nb,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_6223_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = 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))),Nb)) ).

% take_bit_eq_mod
tff(fact_6224_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = 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))),Nb)) ).

% take_bit_eq_mod
tff(fact_6225_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_6226_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_6227_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_6228_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_6229_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_6230_of__bool__odd__eq__mod__2,axiom,
    ! [A: 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(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_6231_of__bool__odd__eq__mod__2,axiom,
    ! [A: 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(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_6232_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_6233_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_6234_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6235_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6236_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6237_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6238_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6239_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6240_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6241_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6242_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6243_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6244_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_6245_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_6246_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6247_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6248_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6249_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6250_take__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_6251_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6252_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: 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,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6253_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6254_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6255_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = 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,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6256_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6257_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6258_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_6259_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6260_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6261_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6262_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_6263_uminus__power__if,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(power_power_int(A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb))) ).

% uminus_power_if
tff(fact_6264_uminus__power__if,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(power_power_real(A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(A),Nb))) ).

% uminus_power_if
tff(fact_6265_uminus__power__if,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(power_power_complex(A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb))) ).

% uminus_power_if
tff(fact_6266_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_6267_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_6268_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_6269_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6270_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6271_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6272_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6273_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6274_realpow__square__minus__le,axiom,
    ! [U: real,X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_6275_ln__one__minus__pos__lower__bound,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X3)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X3))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6276_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Nb,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6277_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6278_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6279_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_6280_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q3)),one_one_int))),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_6281_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => aa_nat_o(P,A3) )
     => ( ! [A3: nat,B3: $o] :
            ( aa_nat_o(P,A3)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_6282_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => aa_int_o(P,A3) )
     => ( ! [A3: int,B3: $o] :
            ( aa_int_o(P,A3)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_6283_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A3 )
         => aa_Code_integer_o(P,A3) )
     => ( ! [A3: code_integer,B3: $o] :
            ( aa_Code_integer_o(P,A3)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A3)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A3 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A3))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_6284_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_se2923211474154528505it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_6285_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_6286_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6287_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6288_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6289_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6290_square__le__1,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_6291_square__le__1,axiom,
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6292_square__le__1,axiom,
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6293_square__le__1,axiom,
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6294_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = 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))),Nb)) ).

% minus_power_mult_self
tff(fact_6295_minus__power__mult__self,axiom,
    ! [A: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb)) = aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6296_minus__power__mult__self,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6297_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6298_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = 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))),Nb)) ).

% minus_power_mult_self
tff(fact_6299_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_6300_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_6301_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_6302_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_6303_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_6304_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_6305_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K2)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6306_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6307_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6308_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_6309_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_6310_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_6311_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_6312_take__bit__int__eq__self,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_6313_take__bit__incr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) != 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))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_6314_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_6315_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_6316_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_6317_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_6318_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_6319_take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_6320_take__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_6321_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),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_Suc
tff(fact_6322_int__bit__induct,axiom,
    ! [P: fun_int_o,K2: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K: int] :
              ( aa_int_o(P,K)
             => ( ( K != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K: int] :
                ( aa_int_o(P,K)
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K2) ) ) ) ) ).

% int_bit_induct
tff(fact_6323_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6324_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6325_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_6326_take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),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,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_6327_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_6328_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Nb,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6329_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Nb: 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,Nb),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))),Nb)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6330_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: 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(Nb),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))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6331_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fw(L)),Qr) ).

% divmod_step_nat_def
tff(fact_6332_ln__one__plus__pos__lower__bound,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(X3),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X3))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6333_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_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))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_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,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_6334_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(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))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_6335_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          aa_o_Code_integer(zero_n356916108424825756nteger,
            ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_6336_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_fx(L)),Qr) ).

% divmod_step_int_def
tff(fact_6337_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_6338_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_6339_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),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,Mb),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,Mb),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_6340_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_6341_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gu),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6342_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gv),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6343_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_gw,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6344_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6345_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_6346_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_6347_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6348_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_6349_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6350_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6351_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6352_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6353_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6354_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6355_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_6356_ComplI,axiom,
    ! [C: complex,A4: set_complex] :
      ( ~ aa_set_complex_o(member_complex(C),A4)
     => aa_set_complex_o(member_complex(C),uminus8566677241136511917omplex(A4)) ) ).

% ComplI
tff(fact_6357_ComplI,axiom,
    ! [C: real,A4: set_real] :
      ( ~ aa_set_real_o(member_real(C),A4)
     => aa_set_real_o(member_real(C),uminus612125837232591019t_real(A4)) ) ).

% ComplI
tff(fact_6358_ComplI,axiom,
    ! [C: $o,A4: set_o] :
      ( ~ aa_set_o_o(member_o((C)),A4)
     => aa_set_o_o(member_o((C)),uminus_uminus_set_o(A4)) ) ).

% ComplI
tff(fact_6359_ComplI,axiom,
    ! [C: nat,A4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C),A4)
     => aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A4)) ) ).

% ComplI
tff(fact_6360_ComplI,axiom,
    ! [C: int,A4: set_int] :
      ( ~ aa_set_int_o(member_int(C),A4)
     => aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A4)) ) ).

% ComplI
tff(fact_6361_Compl__iff,axiom,
    ! [C: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(C),uminus8566677241136511917omplex(A4))
    <=> ~ aa_set_complex_o(member_complex(C),A4) ) ).

% Compl_iff
tff(fact_6362_Compl__iff,axiom,
    ! [C: real,A4: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A4))
    <=> ~ aa_set_real_o(member_real(C),A4) ) ).

% Compl_iff
tff(fact_6363_Compl__iff,axiom,
    ! [C: $o,A4: set_o] :
      ( aa_set_o_o(member_o((C)),uminus_uminus_set_o(A4))
    <=> ~ aa_set_o_o(member_o((C)),A4) ) ).

% Compl_iff
tff(fact_6364_Compl__iff,axiom,
    ! [C: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A4))
    <=> ~ aa_set_nat_o(member_nat(C),A4) ) ).

% Compl_iff
tff(fact_6365_Compl__iff,axiom,
    ! [C: int,A4: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A4))
    <=> ~ aa_set_int_o(member_int(C),A4) ) ).

% Compl_iff
tff(fact_6366_case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,C: fun_se5639961202347202350_nat_o] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P2 = produc2922128104949294807at_nat(A3,B3) )
         => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(C,A3),B3) )
     => aa_Pro5947980084102686253_nat_o(produc410239310623530412_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_6367_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_num_o(aa_num_fun_num_o(C,A3),B3) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_6368_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_num_o(aa_nat_fun_num_o(C,A3),B3) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_6369_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(C,A3),B3) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_6370_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B3) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_6371_case__prodI,axiom,
    ! [F: fun_se5639961202347202350_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(F,A),B)
     => aa_Pro5947980084102686253_nat_o(produc410239310623530412_nat_o(F),produc2922128104949294807at_nat(A,B)) ) ).

% case_prodI
tff(fact_6372_case__prodI,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(F,A),B)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B)) ) ).

% case_prodI
tff(fact_6373_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,A),B)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B)) ) ).

% case_prodI
tff(fact_6374_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_6375_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) ) ).

% case_prodI
tff(fact_6376_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: fun_nu9041852840018610706omplex] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A3),B3)) )
     => aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6377_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A3),B3)) )
     => aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6378_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: $o,C: fun_nu2945343977818489206_set_o] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_set_o_o(member_o((Z)),aa_num_set_o(aa_num_fun_num_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o((Z)),produc1904233663560707350_set_o(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6379_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6380_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6381_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: complex,C: fun_na3925014923085203400omplex] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A3),B3)) )
     => aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6382_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: fun_na3666575854997362246t_real] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A3),B3)) )
     => aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6383_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: $o,C: fun_na3850162164366384704_set_o] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_set_o_o(member_o((Z)),aa_num_set_o(aa_nat_fun_num_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o((Z)),produc836176033315069408_set_o(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6384_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: nat,C: fun_na3896347130304379370et_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6385_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: int,C: fun_na8306170748371236422et_int] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6386_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
      ( aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A),B))
     => aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6387_mem__case__prodI,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
      ( aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6388_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_nu2945343977818489206_set_o,A: num,B: num] :
      ( aa_set_o_o(member_o((Z)),aa_num_set_o(aa_num_fun_num_set_o(C,A),B))
     => aa_set_o_o(member_o((Z)),produc1904233663560707350_set_o(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6389_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6390_mem__case__prodI,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
      ( aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6391_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,A: nat,B: num] :
      ( aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A),B))
     => aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6392_mem__case__prodI,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,A: nat,B: num] :
      ( aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6393_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_na3850162164366384704_set_o,A: nat,B: num] :
      ( aa_set_o_o(member_o((Z)),aa_num_set_o(aa_nat_fun_num_set_o(C,A),B))
     => aa_set_o_o(member_o((Z)),produc836176033315069408_set_o(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6394_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,A: nat,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6395_mem__case__prodI,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,A: nat,B: num] :
      ( aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6396_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,X3: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A3),B3),X3) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),X3) ) ).

% case_prodI2'
tff(fact_6397_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6398_eq__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K2) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_6399_Suc__eq__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K2) )
    <=> ( Nb = aa_num_nat(pred_numeral,K2) ) ) ).

% Suc_eq_numeral
tff(fact_6400_less__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% less_numeral_Suc
tff(fact_6401_less__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% less_Suc_numeral
tff(fact_6402_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2)) ).

% pred_numeral_simps(3)
tff(fact_6403_le__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% le_Suc_numeral
tff(fact_6404_le__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% le_numeral_Suc
tff(fact_6405_diff__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K2)),Nb) ).

% diff_numeral_Suc
tff(fact_6406_diff__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K2)) ).

% diff_Suc_numeral
tff(fact_6407_max__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% max_numeral_Suc
tff(fact_6408_max__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% max_Suc_numeral
tff(fact_6409_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6410_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6411_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6412_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_6413_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_6414_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique3479559517661332726nteger(Mb,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_6415_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_6416_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6417_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_6418_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_6419_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6420_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_6421_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_6422_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gx),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6423_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gy),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6424_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_gz,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6425_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_6426_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_6427_ComplD,axiom,
    ! [C: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(C),uminus8566677241136511917omplex(A4))
     => ~ aa_set_complex_o(member_complex(C),A4) ) ).

% ComplD
tff(fact_6428_ComplD,axiom,
    ! [C: real,A4: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A4))
     => ~ aa_set_real_o(member_real(C),A4) ) ).

% ComplD
tff(fact_6429_ComplD,axiom,
    ! [C: $o,A4: set_o] :
      ( aa_set_o_o(member_o((C)),uminus_uminus_set_o(A4))
     => ~ aa_set_o_o(member_o((C)),A4) ) ).

% ComplD
tff(fact_6430_ComplD,axiom,
    ! [C: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A4))
     => ~ aa_set_nat_o(member_nat(C),A4) ) ).

% ComplD
tff(fact_6431_ComplD,axiom,
    ! [C: int,A4: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A4))
     => ~ aa_set_int_o(member_int(C),A4) ) ).

% ComplD
tff(fact_6432_uminus__set__def,axiom,
    ! [A4: set_real] : uminus612125837232591019t_real(A4) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_a(A4))) ).

% uminus_set_def
tff(fact_6433_uminus__set__def,axiom,
    ! [A4: set_o] : uminus_uminus_set_o(A4) = collect_o(uminus6910356288409981324un_o_o(aTP_Lamm_aa(A4))) ).

% uminus_set_def
tff(fact_6434_uminus__set__def,axiom,
    ! [A4: set_int] : uminus1532241313380277803et_int(A4) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ab(A4))) ).

% uminus_set_def
tff(fact_6435_uminus__set__def,axiom,
    ! [A4: set_nat] : uminus5710092332889474511et_nat(A4) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ac(A4))) ).

% uminus_set_def
tff(fact_6436_uminus__set__def,axiom,
    ! [A4: set_complex] : uminus8566677241136511917omplex(A4) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ad(A4))) ).

% uminus_set_def
tff(fact_6437_uminus__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A4) = collec3392354462482085612at_nat(uminus1662154291547880931_nat_o(aTP_Lamm_ae(A4))) ).

% uminus_set_def
tff(fact_6438_uminus__set__def,axiom,
    ! [A4: set_set_nat] : uminus613421341184616069et_nat(A4) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_af(A4))) ).

% uminus_set_def
tff(fact_6439_uminus__set__def,axiom,
    ! [A4: set_list_nat] : uminus3195874150345416415st_nat(A4) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_ag(A4))) ).

% uminus_set_def
tff(fact_6440_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_ha(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_6441_Collect__neg__eq,axiom,
    ! [P: fun_complex_o] : collect_complex(aTP_Lamm_hb(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_6442_Collect__neg__eq,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] : collec3392354462482085612at_nat(aTP_Lamm_hc(P)) = uminus6524753893492686040at_nat(collec3392354462482085612at_nat(P)) ).

% Collect_neg_eq
tff(fact_6443_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : collect_set_nat(aTP_Lamm_hd(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_6444_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_o] : collect_list_nat(aTP_Lamm_he(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_6445_Compl__eq,axiom,
    ! [A4: set_real] : uminus612125837232591019t_real(A4) = collect_real(aTP_Lamm_hf(A4)) ).

% Compl_eq
tff(fact_6446_Compl__eq,axiom,
    ! [A4: set_o] : uminus_uminus_set_o(A4) = collect_o(aTP_Lamm_hg(A4)) ).

% Compl_eq
tff(fact_6447_Compl__eq,axiom,
    ! [A4: set_int] : uminus1532241313380277803et_int(A4) = collect_int(aTP_Lamm_hh(A4)) ).

% Compl_eq
tff(fact_6448_Compl__eq,axiom,
    ! [A4: set_nat] : uminus5710092332889474511et_nat(A4) = collect_nat(aTP_Lamm_hi(A4)) ).

% Compl_eq
tff(fact_6449_Compl__eq,axiom,
    ! [A4: set_complex] : uminus8566677241136511917omplex(A4) = collect_complex(aTP_Lamm_hj(A4)) ).

% Compl_eq
tff(fact_6450_Compl__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A4) = collec3392354462482085612at_nat(aTP_Lamm_hk(A4)) ).

% Compl_eq
tff(fact_6451_Compl__eq,axiom,
    ! [A4: set_set_nat] : uminus613421341184616069et_nat(A4) = collect_set_nat(aTP_Lamm_hl(A4)) ).

% Compl_eq
tff(fact_6452_Compl__eq,axiom,
    ! [A4: set_list_nat] : uminus3195874150345416415st_nat(A4) = collect_list_nat(aTP_Lamm_hm(A4)) ).

% Compl_eq
tff(fact_6453_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
      ( aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,P2))
     => ~ ! [X5: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X5,Y3) )
           => ~ aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6454_mem__case__prodE,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
      ( aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2))
     => ~ ! [X5: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X5,Y3) )
           => ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6455_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_nu2945343977818489206_set_o,P2: product_prod_num_num] :
      ( aa_set_o_o(member_o((Z)),produc1904233663560707350_set_o(C,P2))
     => ~ ! [X5: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X5,Y3) )
           => ~ aa_set_o_o(member_o((Z)),aa_num_set_o(aa_num_fun_num_set_o(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6456_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
      ( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
     => ~ ! [X5: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X5,Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6457_mem__case__prodE,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
      ( aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2))
     => ~ ! [X5: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X5,Y3) )
           => ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6458_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,P2: product_prod_nat_num] :
      ( aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,P2))
     => ~ ! [X5: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X5,Y3) )
           => ~ aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6459_mem__case__prodE,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
      ( aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2))
     => ~ ! [X5: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X5,Y3) )
           => ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6460_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_na3850162164366384704_set_o,P2: product_prod_nat_num] :
      ( aa_set_o_o(member_o((Z)),produc836176033315069408_set_o(C,P2))
     => ~ ! [X5: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X5,Y3) )
           => ~ aa_set_o_o(member_o((Z)),aa_num_set_o(aa_nat_fun_num_set_o(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6461_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,P2: product_prod_nat_num] :
      ( aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P2))
     => ~ ! [X5: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X5,Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6462_mem__case__prodE,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,P2: product_prod_nat_num] :
      ( aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,P2))
     => ~ ! [X5: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X5,Y3) )
           => ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,X5),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6463_case__prodE,axiom,
    ! [C: fun_se5639961202347202350_nat_o,P2: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(produc410239310623530412_nat_o(C),P2)
     => ~ ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2 = produc2922128104949294807at_nat(X5,Y3) )
           => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(C,X5),Y3) ) ) ).

% case_prodE
tff(fact_6464_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X5: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X5,Y3) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X5),Y3) ) ) ).

% case_prodE
tff(fact_6465_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X5: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X5,Y3) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X5),Y3) ) ) ).

% case_prodE
tff(fact_6466_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X5: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Y3) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X5),Y3) ) ) ).

% case_prodE
tff(fact_6467_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X5),Y3) ) ) ).

% case_prodE
tff(fact_6468_case__prodD,axiom,
    ! [F: fun_se5639961202347202350_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_Pro5947980084102686253_nat_o(produc410239310623530412_nat_o(F),produc2922128104949294807at_nat(A,B))
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_6469_case__prodD,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B))
     => aa_num_o(aa_num_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6470_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B))
     => aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6471_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_6472_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_6473_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Z)
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X5),Y3),Z) ) ) ).

% case_prodE'
tff(fact_6474_case__prodD_H,axiom,
    ! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C) ) ).

% case_prodD'
tff(fact_6475_Collect__case__prod__mono,axiom,
    ! [A4: fun_nat_fun_nat_o,B4: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(A4,B4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A4))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(B4))) ) ).

% Collect_case_prod_mono
tff(fact_6476_numeral__eq__Suc,axiom,
    ! [K2: num] : aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K2)) ).

% numeral_eq_Suc
tff(fact_6477_pred__numeral__def,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),one_one_nat) ).

% pred_numeral_def
tff(fact_6478_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_6479_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_6480_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_6481_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_6482_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_def
tff(fact_6483_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_6484_dbl__dec__def,axiom,
    ! [X3: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X3) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X3),X3)),one_one_complex) ).

% dbl_dec_def
tff(fact_6485_dbl__dec__def,axiom,
    ! [X3: real] : aa_real_real(neg_nu6075765906172075777c_real,X3) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(X3),X3)),one_one_real) ).

% dbl_dec_def
tff(fact_6486_dbl__dec__def,axiom,
    ! [X3: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X3) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),X3)),one_one_rat) ).

% dbl_dec_def
tff(fact_6487_dbl__dec__def,axiom,
    ! [X3: int] : aa_int_int(neg_nu3811975205180677377ec_int,X3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),X3)),one_one_int) ).

% dbl_dec_def
tff(fact_6488_take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_6489_take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_6490_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6491_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6492_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6493_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_6494_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_6495_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_6496_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_6497_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_6498_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X3))),X3))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6499_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_hn),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_6500_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = 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(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_6501_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_6502_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_6503_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_6504_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_6505_split__part,axiom,
    ! [P: $o,Q: fun_nat_fun_nat_o,X: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_ho((P)),Q)),X)
    <=> ( (P)
        & aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Q),X) ) ) ).

% split_part
tff(fact_6506_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6507_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_6508_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_6509_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_6510_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6511_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_6512_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_6513_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_6514_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6515_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6516_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6517_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6518_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% abs_numeral
tff(fact_6519_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_6520_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_6521_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_6522_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_6523_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6524_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6525_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_6526_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_6527_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_6528_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_6529_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_6530_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_6531_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_6532_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_6533_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_6534_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_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_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_6543_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_6544_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_6545_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_6546_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_6547_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_6548_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% abs_neg_numeral
tff(fact_6549_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_6550_abs__power__minus,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) ).

% abs_power_minus
tff(fact_6551_abs__power__minus,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),Nb)) ).

% abs_power_minus
tff(fact_6552_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_6553_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_6554_pred__numeral__inc,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% pred_numeral_inc
tff(fact_6555_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6556_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6557_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6558_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6559_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_6560_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_6561_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_6562_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_6563_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q3),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_6564_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6565_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6566_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6567_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6568_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6569_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6570_power2__abs,axiom,
    ! [A: real] : aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6571_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6572_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6573_abs__power2,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6574_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6575_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6576_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6577_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6578_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6579_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6580_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6581_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6582_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6583_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6584_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6585_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6586_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6587_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6588_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6589_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6590_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6591_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6592_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6593_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6594_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6595_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6596_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6597_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6598_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6599_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6600_numeral__div__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(Mb,Nb))) ).

% numeral_div_minus_numeral
tff(fact_6601_minus__numeral__div__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_div_numeral
tff(fact_6602_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aTP_Lamm_hp),Prod) ).

% prod.disc_eq_case
tff(fact_6603_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_6604_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_6605_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_6606_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_6607_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_6608_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_6609_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_6610_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_6611_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_6612_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_6613_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)) ).

% abs_minus_commute
tff(fact_6614_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_6615_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_6616_power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) ).

% power_abs
tff(fact_6617_power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),Nb)) = aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb) ).

% power_abs
tff(fact_6618_power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) ).

% power_abs
tff(fact_6619_num__induct,axiom,
    ! [P: fun_num_o,X3: num] :
      ( aa_num_o(P,one)
     => ( ! [X5: num] :
            ( aa_num_o(P,X5)
           => aa_num_o(P,aa_num_num(inc,X5)) )
       => aa_num_o(P,X3) ) ) ).

% num_induct
tff(fact_6620_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_6621_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_6622_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_6623_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_6624_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_6625_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_6626_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_6627_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_6628_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_6629_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_6630_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_6631_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_6632_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_6633_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_6634_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6635_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_6636_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_6637_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6638_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6639_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6640_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_6641_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq3
tff(fact_6642_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq3
tff(fact_6643_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_6644_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_6645_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq2
tff(fact_6646_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq2
tff(fact_6647_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_6648_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_6649_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_6650_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_6651_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_6652_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_6653_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_6654_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_6655_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_6656_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_6657_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_6658_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_6659_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_6660_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_6661_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_6662_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_6663_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_6664_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_6665_inc_Osimps_I2_J,axiom,
    ! [X3: num] : aa_num_num(inc,aa_num_num(bit0,X3)) = aa_num_num(bit1,X3) ).

% inc.simps(2)
tff(fact_6666_inc_Osimps_I3_J,axiom,
    ! [X3: num] : aa_num_num(inc,aa_num_num(bit1,X3)) = aa_num_num(bit0,aa_num_num(inc,X3)) ).

% inc.simps(3)
tff(fact_6667_dense__eq0__I,axiom,
    ! [X3: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),E2) )
     => ( X3 = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6668_dense__eq0__I,axiom,
    ! [X3: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X3)),E2) )
     => ( X3 = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6669_abs__mult__pos,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X3) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_6670_abs__mult__pos,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y)),X3) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_6671_abs__mult__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X3) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_6672_abs__mult__pos,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X3) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_6673_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_6674_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6675_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6676_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_6677_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6678_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6679_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6680_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6681_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6682_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6683_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6684_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6685_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_6686_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_6687_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_6688_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6689_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_6690_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6691_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6692_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6693_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_6694_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_6695_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_6696_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_6697_abs__diff__le__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X3,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),X3)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6698_abs__diff__le__iff,axiom,
    ! [X3: 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(minus_minus_real(X3),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),R2)),X3)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(plus_plus_real(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6699_abs__diff__le__iff,axiom,
    ! [X3: 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,X3),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)),X3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6700_abs__diff__le__iff,axiom,
    ! [X3: 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,X3),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)),X3)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6701_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6702_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6703_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6704_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6705_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_6706_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6707_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6708_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_6709_abs__diff__less__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X3,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),X3)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6710_abs__diff__less__iff,axiom,
    ! [X3: 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(minus_minus_real(X3),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),R2)),X3)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(plus_plus_real(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6711_abs__diff__less__iff,axiom,
    ! [X3: 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,X3),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)),X3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6712_abs__diff__less__iff,axiom,
    ! [X3: 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,X3),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)),X3)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6713_add__One,axiom,
    ! [X3: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X3),one) = aa_num_num(inc,X3) ).

% add_One
tff(fact_6714_abs__add__one__gt__zero,axiom,
    ! [X3: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X3))) ).

% abs_add_one_gt_zero
tff(fact_6715_abs__add__one__gt__zero,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,X3))) ).

% abs_add_one_gt_zero
tff(fact_6716_abs__add__one__gt__zero,axiom,
    ! [X3: 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,X3))) ).

% abs_add_one_gt_zero
tff(fact_6717_abs__add__one__gt__zero,axiom,
    ! [X3: 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,X3))) ).

% abs_add_one_gt_zero
tff(fact_6718_numeral__inc,axiom,
    ! [X3: num] : numera6690914467698888265omplex(aa_num_num(inc,X3)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X3)),one_one_complex) ).

% numeral_inc
tff(fact_6719_numeral__inc,axiom,
    ! [X3: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,X3)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,X3)),one_one_real) ).

% numeral_inc
tff(fact_6720_numeral__inc,axiom,
    ! [X3: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,X3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X3)),one_one_rat) ).

% numeral_inc
tff(fact_6721_numeral__inc,axiom,
    ! [X3: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,X3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X3)),one_one_nat) ).

% numeral_inc
tff(fact_6722_numeral__inc,axiom,
    ! [X3: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,X3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X3)),one_one_int) ).

% numeral_inc
tff(fact_6723_abs__le__square__iff,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X3)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6724_abs__le__square__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6725_abs__le__square__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X3)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6726_abs__le__square__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X3)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6727_abs__square__eq__1,axiom,
    ! [X3: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X3) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6728_abs__square__eq__1,axiom,
    ! [X3: rat] :
      ( ( aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,X3) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6729_abs__square__eq__1,axiom,
    ! [X3: real] :
      ( ( aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,X3) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6730_abs__square__eq__1,axiom,
    ! [X3: int] :
      ( ( aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X3) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6731_power__even__abs,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_6732_power__even__abs,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_even_abs
tff(fact_6733_power__even__abs,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(power_power_real(A),Nb) ) ) ).

% power_even_abs
tff(fact_6734_power__even__abs,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_even_abs
tff(fact_6735_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X3)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6736_power2__le__iff__abs__le,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6737_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X3)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6738_power2__le__iff__abs__le,axiom,
    ! [Y: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X3)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6739_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,X3: code_integer] :
      ( ! [X5: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X5),aa_nat_Code_integer(power_8256067586552552935nteger(X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(X3)),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6740_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,X3: real] :
      ( ! [X5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X5)
         => aa_real_o(aa_real_fun_real_o(P,X5),aa_nat_real(power_power_real(X5),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,X3)),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6741_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,X3: rat] :
      ( ! [X5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X5)
         => aa_rat_o(aa_rat_fun_rat_o(P,X5),aa_nat_rat(power_power_rat(X5),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,X3)),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6742_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,X3: int] :
      ( ! [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => aa_int_o(aa_int_fun_int_o(P,X5),aa_nat_int(power_power_int(X5),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,X3)),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6743_abs__square__le__1,axiom,
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X3),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(X3)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6744_abs__square__le__1,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),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,X3)),one_one_real) ) ).

% abs_square_le_1
tff(fact_6745_abs__square__le__1,axiom,
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X3),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,X3)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6746_abs__square__le__1,axiom,
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X3),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,X3)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6747_abs__square__less__1,axiom,
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X3),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(X3)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6748_abs__square__less__1,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X3),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,X3)),one_one_real) ) ).

% abs_square_less_1
tff(fact_6749_abs__square__less__1,axiom,
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X3),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,X3)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6750_abs__square__less__1,axiom,
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X3),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,X3)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6751_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6752_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6753_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6754_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6755_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_6756_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X3))),X3))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6757_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X3))),X3))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6758_tanh__ln__real,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X3)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6759_and__int_Oelims,axiom,
    ! [X3: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Xa2) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(X3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2) ))),
            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))),X3)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_6760_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_6761_arctan__double,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X3)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_6762_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_6763_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_6764_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_6765_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6766_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_6767_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6768_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6769_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6770_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6771_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_6772_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_6773_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_6774_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_6775_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ).

% and_numerals(2)
tff(fact_6776_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_6777_and__numerals_I8_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_6778_and__numerals_I8_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_6779_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K2)) ).

% pred_numeral_simps(2)
tff(fact_6780_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_dec_simps(1)
tff(fact_6781_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_dec_simps(1)
tff(fact_6782_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2))) ).

% dbl_dec_simps(1)
tff(fact_6783_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K2))) ).

% dbl_dec_simps(1)
tff(fact_6784_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_dec_simps(1)
tff(fact_6785_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_inc_simps(1)
tff(fact_6786_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_inc_simps(1)
tff(fact_6787_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2))) ).

% dbl_inc_simps(1)
tff(fact_6788_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(aa_num_Code_integer(numera6620942414471956472nteger,K2))) ).

% dbl_inc_simps(1)
tff(fact_6789_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_inc_simps(1)
tff(fact_6790_and__numerals_I5_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X3))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6791_and__numerals_I5_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6792_and__numerals_I1_J,axiom,
    ! [Y: 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,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_6793_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_6794_and__numerals_I3_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_6795_and__numerals_I3_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(3)
tff(fact_6796_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_6797_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_6798_and__numerals_I6_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_6799_and__numerals_I6_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(6)
tff(fact_6800_and__numerals_I4_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_6801_and__numerals_I4_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(4)
tff(fact_6802_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_6803_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_6804_and__numerals_I7_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_6805_and__numerals_I7_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_6806_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_6807_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_6808_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_6809_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_6810_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_6811_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6812_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6813_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6814_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_6815_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_6816_even__and__iff__int,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_6817_dbl__inc__def,axiom,
    ! [X3: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X3),X3)),one_one_complex) ).

% dbl_inc_def
tff(fact_6818_dbl__inc__def,axiom,
    ! [X3: real] : aa_real_real(neg_nu8295874005876285629c_real,X3) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(X3),X3)),one_one_real) ).

% dbl_inc_def
tff(fact_6819_dbl__inc__def,axiom,
    ! [X3: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),X3)),one_one_rat) ).

% dbl_inc_def
tff(fact_6820_dbl__inc__def,axiom,
    ! [X3: int] : aa_int_int(neg_nu5851722552734809277nc_int,X3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),X3)),one_one_int) ).

% dbl_inc_def
tff(fact_6821_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6822_and__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),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))) ).

% and_one_eq
tff(fact_6823_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6824_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6825_one__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_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))) ).

% one_and_eq
tff(fact_6826_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6827_numeral__BitM,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bitM,Nb)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).

% numeral_BitM
tff(fact_6828_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,Nb)) = aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb))),one_one_real) ).

% numeral_BitM
tff(fact_6829_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb))),one_one_rat) ).

% numeral_BitM
tff(fact_6830_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).

% numeral_BitM
tff(fact_6831_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_6832_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_6833_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_6834_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K2)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_abs_add_iff
tff(fact_6835_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_add_abs_iff
tff(fact_6836_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
                & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6837_and__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_6838_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6839_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6840_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K2 = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_6841_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
            ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_6842_and__int_Opelims,axiom,
    ! [X3: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Xa2) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(X3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2) ))),
                  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))),X3)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_6843_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_int(ring_1_of_int_int,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_6844_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_real(ring_1_of_int_real,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: real,
            l:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(plus_plus_real(l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_6845_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_complex(ring_17405671764205052669omplex,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_6846_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K2) = $ite(
        K2 = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_6847_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_rat(ring_1_of_int_rat,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: rat,
            l:= aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_6848_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_6849_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_6850_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_6851_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_6852_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6853_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6854_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6855_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6856_of__int__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K2)) = numera6690914467698888265omplex(K2) ).

% of_int_numeral
tff(fact_6857_of__int__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K2)) = aa_num_real(numeral_numeral_real,K2) ).

% of_int_numeral
tff(fact_6858_of__int__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K2)) = aa_num_rat(numeral_numeral_rat,K2) ).

% of_int_numeral
tff(fact_6859_of__int__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,K2) ).

% of_int_numeral
tff(fact_6860_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_6861_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_6862_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_6863_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_6864_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_6865_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_6866_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).

% of_int_power
tff(fact_6867_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_6868_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_6869_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_6870_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,X3) )
    <=> ( aa_nat_int(power_power_int(B),W) = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6871_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,X3) )
    <=> ( aa_nat_int(power_power_int(B),W) = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6872_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X3) )
    <=> ( aa_nat_int(power_power_int(B),W) = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6873_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,X3) )
    <=> ( aa_nat_int(power_power_int(B),W) = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6874_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X3) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( X3 = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6875_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X3) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( X3 = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6876_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X3) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( X3 = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6877_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X3) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( X3 = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6878_and__nat__numerals_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_6879_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_6880_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6881_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6882_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6883_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6884_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_6885_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_6886_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_6887_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_6888_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6889_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_6890_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_6891_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6892_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6893_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6894_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6895_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6896_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6897_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_6898_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_6899_and__nat__numerals_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_6900_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6901_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6902_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_6903_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6904_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6905_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6906_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_6907_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_6908_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6909_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_6910_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_6911_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6912_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X3)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6913_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6914_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X3)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6915_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6916_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X3)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6917_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6918_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X3)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6919_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6920_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6921_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6922_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X3)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6923_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X3) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6924_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X3) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6925_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X3) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6926_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),X3) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6927_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),X3) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6928_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),X3) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6929_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,X3)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X3),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6930_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X3)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X3),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6931_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,X3)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X3),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6932_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_6933_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_6934_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_6935_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_6936_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_6937_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6938_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_6939_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_6940_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_6941_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_6942_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_6943_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_6944_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_6945_bit__mod__2__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6946_bit__mod__2__iff,axiom,
    ! [A: int,Nb: 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)))),Nb)
    <=> ( ( Nb = 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_6947_bit__mod__2__iff,axiom,
    ! [A: nat,Nb: 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)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6948_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb))
    <=> 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,X3)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6949_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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,X3)),Nb))
    <=> 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,X3)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6950_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb))
    <=> 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,X3)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6951_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb)),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,X3)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6952_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3)),Nb)),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,X3)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6953_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3)),Nb)),aa_int_int(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,X3)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6954_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6955_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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,X3)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6956_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6957_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb)),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,X3)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6958_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3)),Nb)),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,X3)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6959_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3)),Nb)),aa_int_int(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,X3)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6960_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6961_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X3))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6962_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X3))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6963_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X3))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6964_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X3))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6965_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6966_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X3))),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6967_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X3))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6968_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X3))),Nb) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6969_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X3))),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6970_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6971_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6972_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6973_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6974_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X3))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6975_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X3))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6976_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6977_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6978_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6979_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X3))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6980_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X3))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6981_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6982_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6983_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6984_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6985_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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,X3))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X3))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6986_bit__numeral__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_iff
tff(fact_6987_bit__numeral__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_iff
tff(fact_6988_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6989_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6990_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_6991_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_6992_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_6993_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_6994_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_6995_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_6996_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_6997_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_6998_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_6999_of__int__leD,axiom,
    ! [Nb: int,X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),X3)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X3) ) ) ).

% of_int_leD
tff(fact_7000_of__int__leD,axiom,
    ! [Nb: int,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),X3)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X3) ) ) ).

% of_int_leD
tff(fact_7001_of__int__leD,axiom,
    ! [Nb: int,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),X3)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X3) ) ) ).

% of_int_leD
tff(fact_7002_of__int__leD,axiom,
    ! [Nb: int,X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),X3)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X3) ) ) ).

% of_int_leD
tff(fact_7003_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)) ).

% of_int_neg_numeral
tff(fact_7004_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2)) ).

% of_int_neg_numeral
tff(fact_7005_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2)) ).

% of_int_neg_numeral
tff(fact_7006_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K2)) ).

% of_int_neg_numeral
tff(fact_7007_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2)) ).

% of_int_neg_numeral
tff(fact_7008_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),K2),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_7009_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7010_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: 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))),Nb) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7011_bit__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,Nb))
    <=> 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)))),Nb) ) ).

% bit_Suc
tff(fact_7012_bit__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Nb))
    <=> 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)))),Nb) ) ).

% bit_Suc
tff(fact_7013_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_7014_stable__imp__bit__iff__odd,axiom,
    ! [A: int,Nb: 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_nat_o(bit_se1146084159140164899it_int(A),Nb)
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_7015_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,Nb: 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_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_7016_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) )
     => ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_7017_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(A),N)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_7018_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_7019_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K2),M2)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K2),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),N) ) ) ) ).

% int_bit_bound
tff(fact_7020_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_7021_bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_7022_bit__iff__odd,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_7023_bit__iff__odd,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_7024_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Nb: 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))),Nb)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7025_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Nb: 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))),Nb)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7026_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_7027_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_7028_even__of__int__iff,axiom,
    ! [K2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_of_int_iff
tff(fact_7029_even__of__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_of_int_iff
tff(fact_7030_bit__int__def,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_7031_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7032_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7033_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7034_even__bit__succ__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A)),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7035_even__bit__succ__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7036_even__bit__succ__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7037_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7038_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Nb: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7039_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
      <=> ( 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)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7040_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_7041_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_7042_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_7043_set__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_7044_unset__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_7045_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7046_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,Nb: 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))),B))),Nb)
      <=> $ite(Nb = 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))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7047_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,Nb: 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))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7048_bit__rec,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_7049_bit__rec,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> $ite(Nb = 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,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_7050_bit__rec,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_7051_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_7052_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_7053_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_7054_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_7055_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L3))
           => ( ( ~ ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_7056_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% take_bit_Suc_from_most
tff(fact_7057_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_7058_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_7059_floor__exists1,axiom,
    ! [X3: real] :
    ? [X5: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X5)),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y5)),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
tff(fact_7060_floor__exists1,axiom,
    ! [X3: rat] :
    ? [X5: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X5)),X3)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y5)),X3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
tff(fact_7061_floor__exists,axiom,
    ! [X3: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),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_7062_floor__exists,axiom,
    ! [X3: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X3)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),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_7063_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I3: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_7064_tanh__real__altdef,axiom,
    ! [X3: real] : aa_real_real(tanh_real,X3) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)))) ).

% tanh_real_altdef
tff(fact_7065_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K2)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(pow,K2),L)) ).

% power_numeral
tff(fact_7066_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(pow,K2),L)) ).

% power_numeral
tff(fact_7067_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(pow,K2),L)) ).

% power_numeral
tff(fact_7068_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(pow,K2),L)) ).

% power_numeral
tff(fact_7069_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(pow,K2),L)) ).

% power_numeral
tff(fact_7070_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = $ite(
        ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K2 = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_7071_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% or_numerals(2)
tff(fact_7072_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_numerals(2)
tff(fact_7073_or__numerals_I8_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3)) ).

% or_numerals(8)
tff(fact_7074_or__numerals_I8_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% or_numerals(8)
tff(fact_7075_or__numerals_I3_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y))) ).

% or_numerals(3)
tff(fact_7076_or__numerals_I3_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = 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_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y))) ).

% or_numerals(3)
tff(fact_7077_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% or_numerals(1)
tff(fact_7078_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_numerals(1)
tff(fact_7079_or__numerals_I5_J,axiom,
    ! [X3: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X3))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3)) ).

% or_numerals(5)
tff(fact_7080_or__numerals_I5_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% or_numerals(5)
tff(fact_7081_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_7082_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_7083_or__numerals_I7_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(7)
tff(fact_7084_or__numerals_I7_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(7)
tff(fact_7085_or__numerals_I6_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(6)
tff(fact_7086_or__numerals_I6_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(6)
tff(fact_7087_or__numerals_I4_J,axiom,
    ! [X3: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X3))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X3)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(4)
tff(fact_7088_or__numerals_I4_J,axiom,
    ! [X3: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X3)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(4)
tff(fact_7089_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_7090_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_7091_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) ) ) ).

% disjunctive_add
tff(fact_7092_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ) ) ).

% disjunctive_add
tff(fact_7093_exp__add__commuting,axiom,
    ! [X3: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(X3),Y) = aa_complex_complex(times_times_complex(Y),X3) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X3),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X3)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_7094_exp__add__commuting,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(times_times_real(X3),Y) = aa_real_real(times_times_real(Y),X3) )
     => ( aa_real_real(exp_real,aa_real_real(plus_plus_real(X3),Y)) = aa_real_real(times_times_real(aa_real_real(exp_real,X3)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_7095_mult__exp__exp,axiom,
    ! [X3: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X3)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X3),Y)) ).

% mult_exp_exp
tff(fact_7096_mult__exp__exp,axiom,
    ! [X3: real,Y: real] : aa_real_real(times_times_real(aa_real_real(exp_real,X3)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(plus_plus_real(X3),Y)) ).

% mult_exp_exp
tff(fact_7097_pow_Osimps_I1_J,axiom,
    ! [X3: num] : aa_num_num(aa_num_fun_num_num(pow,X3),one) = X3 ).

% pow.simps(1)
tff(fact_7098_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_7099_even__or__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1080825931792720795nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_7100_even__or__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_7101_even__or__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_7102_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_7103_tanh__altdef,axiom,
    ! [X3: real] : aa_real_real(tanh_real,X3) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X3)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X3)))),aa_real_real(plus_plus_real(aa_real_real(exp_real,X3)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X3)))) ).

% tanh_altdef
tff(fact_7104_tanh__altdef,axiom,
    ! [X3: complex] : aa_complex_complex(tanh_complex,X3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X3)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X3)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X3)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X3)))) ).

% tanh_altdef
tff(fact_7105_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_7106_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_7107_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% mask_Suc_exp
tff(fact_7108_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_Suc_exp
tff(fact_7109_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_7110_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)) = aa_nat_real(power_power_real(aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_7111_one__or__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(one_one_Code_integer,A) = plus_p5714425477246183910nteger(A,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_7112_one__or__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),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))) ).

% one_or_eq
tff(fact_7113_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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))) ).

% one_or_eq
tff(fact_7114_or__one__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(A,one_one_Code_integer) = plus_p5714425477246183910nteger(A,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_7115_or__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),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))) ).

% or_one_eq
tff(fact_7116_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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))) ).

% or_one_eq
tff(fact_7117_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))) ).

% mask_Suc_double
tff(fact_7118_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))) ).

% mask_Suc_double
tff(fact_7119_OR__upper,axiom,
    ! [X3: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X3),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_7120_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_1: nat] : aa_nat_o(P,X_1)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_7121_ex__le__of__int,axiom,
    ! [X3: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_7122_ex__le__of__int,axiom,
    ! [X3: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_7123_exp__bound,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X3)),aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X3)),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_7124_or__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_7125_real__exp__bound__lemma,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X3)),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3))) ) ) ).

% real_exp_bound_lemma
tff(fact_7126_exp__lower__Taylor__quadratic,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X3)),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(X3),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,X3)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7127_round__unique,axiom,
    ! [X3: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,X3) = Y ) ) ) ).

% round_unique
tff(fact_7128_round__unique,axiom,
    ! [X3: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),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,X3) = Y ) ) ) ).

% round_unique
tff(fact_7129_round__unique_H,axiom,
    ! [X3: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,X3) = Nb ) ) ).

% round_unique'
tff(fact_7130_round__unique_H,axiom,
    ! [X3: rat,Nb: 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,X3),aa_int_rat(ring_1_of_int_rat,Nb)))),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,X3) = Nb ) ) ).

% round_unique'
tff(fact_7131_of__int__round__abs__le,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X3))),X3))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7132_of__int__round__abs__le,axiom,
    ! [X3: 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,X3))),X3))),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_7133_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_7134_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_7135_of__int__round__gt,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X3))) ).

% of_int_round_gt
tff(fact_7136_of__int__round__gt,axiom,
    ! [X3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),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,X3))) ).

% of_int_round_gt
tff(fact_7137_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_7138_round__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_7139_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_7140_or__nat__numerals_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% or_nat_numerals(4)
tff(fact_7141_or__nat__numerals_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% or_nat_numerals(3)
tff(fact_7142_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_7143_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7144_round__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7145_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_7146_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_7147_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_7148_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_7149_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_7150_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_7151_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_7152_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_7153_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_7154_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_7155_round__mono,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X3)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_7156_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_7157_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_7158_or__not__num__neg_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X3),Xa2) = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X3 = one )
           => ! [M: num] :
                ( ( Xa2 = aa_num_num(bit0,M) )
               => ( Y != aa_num_num(bit1,M) ) ) )
         => ( ( ( X3 = one )
             => ! [M: num] :
                  ( ( Xa2 = aa_num_num(bit1,M) )
                 => ( Y != aa_num_num(bit1,M) ) ) )
           => ( ( ? [N: num] : X3 = aa_num_num(bit0,N)
               => ( ( Xa2 = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( X3 = aa_num_num(bit0,N) )
                   => ! [M: num] :
                        ( ( Xa2 = aa_num_num(bit0,M) )
                       => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
               => ( ! [N: num] :
                      ( ( X3 = aa_num_num(bit0,N) )
                     => ! [M: num] :
                          ( ( Xa2 = aa_num_num(bit1,M) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
                 => ( ( ? [N: num] : X3 = aa_num_num(bit1,N)
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( X3 = aa_num_num(bit1,N) )
                         => ! [M: num] :
                              ( ( Xa2 = aa_num_num(bit0,M) )
                             => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X3 = aa_num_num(bit1,N) )
                           => ! [M: num] :
                                ( ( Xa2 = aa_num_num(bit1,M) )
                               => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_7159_round__diff__minimal,axiom,
    ! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_7160_round__diff__minimal,axiom,
    ! [Z: rat,Mb: 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,Mb)))) ).

% round_diff_minimal
tff(fact_7161_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_7162_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_7163_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_7164_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_7165_of__int__round__le,axiom,
    ! [X3: 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,X3))),aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7166_of__int__round__le,axiom,
    ! [X3: 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,X3))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),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_7167_of__int__round__ge,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X3))) ).

% of_int_round_ge
tff(fact_7168_of__int__round__ge,axiom,
    ! [X3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),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,X3))) ).

% of_int_round_ge
tff(fact_7169_log__base__10__eq1,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X3) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X3)) ) ) ).

% log_base_10_eq1
tff(fact_7170_arctan__half,axiom,
    ! [X3: real] : aa_real_real(arctan,X3) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(X3),aa_real_real(plus_plus_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_7171_log__base__10__eq2,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X3) = aa_real_real(times_times_real(aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X3)) ) ) ).

% log_base_10_eq2
tff(fact_7172_machin,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_7173_machin__Euler,axiom,
    aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_7174_sqrt__sum__squares__half__less,axiom,
    ! [X3: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_7175_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_7176_real__sqrt__abs,axiom,
    ! [X3: real] : aa_real_real(sqrt,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X3) ).

% real_sqrt_abs
tff(fact_7177_real__sqrt__pow2,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X3 ) ) ).

% real_sqrt_pow2
tff(fact_7178_real__sqrt__pow2__iff,axiom,
    ! [X3: real] :
      ( ( aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X3 )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3) ) ).

% real_sqrt_pow2_iff
tff(fact_7179_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X3: real,Y: real,Xa2: real,Ya: real] : aa_nat_real(power_power_real(aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_7180_real__sqrt__power,axiom,
    ! [X3: real,K2: nat] : aa_real_real(sqrt,aa_nat_real(power_power_real(X3),K2)) = aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),K2) ).

% real_sqrt_power
tff(fact_7181_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_7182_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_7183_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_7184_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_7185_real__less__rsqrt,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_7186_sqrt__le__D,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X3)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_7187_real__le__rsqrt,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_7188_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7189_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_7190_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_7191_real__le__lsqrt,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X3)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_7192_real__sqrt__unique,axiom,
    ! [Y: real,X3: real] :
      ( ( aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X3 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,X3) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_7193_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7194_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X3 )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_7195_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( X3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_7196_real__sqrt__sum__squares__ge1,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_7197_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_7198_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_7199_sqrt__ge__absD,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_7200_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_7201_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_7202_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_7203_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_7204_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_7205_real__less__lsqrt,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X3)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_7206_sqrt__sum__squares__le__sum,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(X3),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_7207_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_7208_real__sqrt__ge__abs1,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_7209_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_7210_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,X3)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_7211_ln__sqrt,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X3)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_7212_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_7213_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_7214_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_7215_arsinh__real__aux,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(X3),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_7216_real__sqrt__power__even,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
       => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),Nb) = aa_nat_real(power_power_real(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_7217_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X3: real,Y: real,Xa2: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_7218_arith__geo__mean__sqrt,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(X3),Y))),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_7219_cos__x__y__le__one,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X3),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_7220_real__sqrt__sum__squares__less,axiom,
    ! [X3: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_7221_arcosh__real__def,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X3)
     => ( aa_real_real(arcosh_real,X3) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X3),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_7222_arsinh__real__def,axiom,
    ! [X3: real] : aa_real_real(arsinh_real,X3) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X3),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_7223_cot__less__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,X3)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_7224_cot__periodic,axiom,
    ! [X3: real] : aa_real_real(cot_real,aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X3) ).

% cot_periodic
tff(fact_7225_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7226_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_7227_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7228_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7229_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7230_sin__periodic,axiom,
    ! [X3: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X3) ).

% sin_periodic
tff(fact_7231_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_7232_sin__2pi__minus,axiom,
    ! [X3: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X3)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X3)) ).

% sin_2pi_minus
tff(fact_7233_arcsin__sin,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X3)) = X3 ) ) ) ).

% arcsin_sin
tff(fact_7234_le__arcsin__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,X3))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),X3) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_7235_arcsin__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X3)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_7236_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_7237_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_7238_sin__gt__zero__02,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_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,X3)) ) ) ).

% sin_gt_zero_02
tff(fact_7239_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7240_sin__gt__zero2,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(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,X3)) ) ) ).

% sin_gt_zero2
tff(fact_7241_sin__lt__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X3)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_7242_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_7243_sin__monotone__2pi__le,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X3)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7244_sin__mono__le__eq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7245_sin__inj__pi,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,X3) = aa_real_real(sin_real,Y) )
             => ( X3 = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7246_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_7247_sin__le__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X3)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_7248_sin__less__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X3)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_7249_sin__mono__less__eq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7250_sin__monotone__2pi,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X3)) ) ) ) ).

% sin_monotone_2pi
tff(fact_7251_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X5) = Y )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y5) = Y ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% sin_total
tff(fact_7252_sin__zero__iff__int,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sin_real,X3) = zero_zero_real )
    <=> ? [I4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4)
          & ( X3 = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7253_sin__arctan,axiom,
    ! [X3: real] : aa_real_real(sin_real,aa_real_real(arctan,X3)) = aa_real_real(divide_divide_real(X3),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7254_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_7255_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_7256_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_7257_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_7258_cot__gt__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(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,X3)) ) ) ).

% cot_gt_zero
tff(fact_7259_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_7260_sincos__total__2pi,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( X3 = aa_real_real(cos_real,T4) )
               => ( Y != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7261_sin__tan,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,X3) = aa_real_real(divide_divide_real(aa_real_real(tan_real,X3)),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7262_sincos__total__2pi__le,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( X3 = aa_real_real(cos_real,T4) )
          & ( Y = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7263_sincos__total__pi__half,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( X3 = aa_real_real(cos_real,T4) )
              & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7264_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_7265_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7266_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7267_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7268_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_7269_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_7270_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_7271_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_7272_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7273_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7274_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7275_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7276_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7277_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7278_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7279_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7280_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_7281_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_7282_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_7283_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_7284_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7285_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7286_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7287_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7288_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_7289_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_7290_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_7291_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_7292_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_7293_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7294_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7295_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7296_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7297_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_7298_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_7299_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_7300_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_7301_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_7302_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_7303_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_7304_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_7305_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7306_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7307_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7308_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7309_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7310_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7311_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7312_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7313_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7314_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7315_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_7316_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7317_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_7318_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7319_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7320_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).

% of_nat_power
tff(fact_7321_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).

% of_nat_power
tff(fact_7322_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).

% of_nat_power
tff(fact_7323_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).

% of_nat_power
tff(fact_7324_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,X3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7325_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,X3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7326_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7327_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7328_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,X3) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( X3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7329_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X3) = aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( X3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7330_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,X3) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( X3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7331_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,X3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( X3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7332_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_7333_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_7334_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_7335_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_Code_integer(zero_n356916108424825756nteger,(P)) ).

% of_nat_of_bool
tff(fact_7336_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7337_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7338_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7339_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7340_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_7341_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_7342_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_7343_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_7344_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_7345_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7346_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7347_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7348_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7349_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7350_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7351_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7352_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7353_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7354_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7355_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7356_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7357_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X3)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7358_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X3)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7359_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7360_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7361_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7362_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X3)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7363_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X3)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7364_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7365_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7366_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7367_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7368_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7369_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7370_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X3) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7371_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7372_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7373_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7374_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7375_sin__cos__squared__add3,axiom,
    ! [X3: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X3)),aa_complex_complex(cos_complex,X3))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X3)),aa_complex_complex(sin_complex,X3))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_7376_sin__cos__squared__add3,axiom,
    ! [X3: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X3)),aa_real_real(cos_real,X3))),aa_real_real(times_times_real(aa_real_real(sin_real,X3)),aa_real_real(sin_real,X3))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7377_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_7378_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_7379_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7380_tan__periodic__n,axiom,
    ! [X3: real,Nb: num] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,X3) ).

% tan_periodic_n
tff(fact_7381_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X3)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X3)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7382_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X3)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X3)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7383_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X3)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X3)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7384_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X3)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7385_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_nat_real(power_power_real(A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_7386_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7387_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7388_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7389_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7390_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7391_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7392_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7393_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7394_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7395_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7396_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7397_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7398_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7399_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7400_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,X3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,X3))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),X3) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7401_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7402_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7403_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7404_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7405_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7406_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7407_cos__periodic,axiom,
    ! [X3: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X3) ).

% cos_periodic
tff(fact_7408_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_7409_cos__2pi__minus,axiom,
    ! [X3: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X3)) = aa_real_real(cos_real,X3) ).

% cos_2pi_minus
tff(fact_7410_tan__periodic,axiom,
    ! [X3: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X3) ).

% tan_periodic
tff(fact_7411_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_7412_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_7413_sin__cos__squared__add,axiom,
    ! [X3: real] : aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(sin_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7414_sin__cos__squared__add,axiom,
    ! [X3: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7415_sin__cos__squared__add2,axiom,
    ! [X3: real] : aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7416_sin__cos__squared__add2,axiom,
    ! [X3: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7417_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7418_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7419_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_7420_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7421_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_7422_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7423_real__arch__simple,axiom,
    ! [X3: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_7424_real__arch__simple,axiom,
    ! [X3: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_7425_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X3)) ).

% mult_of_nat_commute
tff(fact_7426_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X3)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(semiri5074537144036343181t_real,X3)) ).

% mult_of_nat_commute
tff(fact_7427_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X3)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X3)) ).

% mult_of_nat_commute
tff(fact_7428_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X3)) ).

% mult_of_nat_commute
tff(fact_7429_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_7430_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_7431_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_7432_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_7433_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_7434_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_7435_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_7436_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_7437_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_7438_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_7439_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_7440_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_7441_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7442_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7443_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7444_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7445_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7446_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7447_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7448_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7449_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_7450_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_7451_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_7452_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_7453_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_7454_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_7455_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_7456_add__tan__eq,axiom,
    ! [X3: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X3) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X3)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X3),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X3)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7457_add__tan__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(cos_real,X3) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(X3),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7458_sin__add,axiom,
    ! [X3: real,Y: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X3),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,X3)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,X3)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_7459_lemma__tan__add1,axiom,
    ! [X3: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X3) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X3)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X3),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X3)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7460_lemma__tan__add1,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(cos_real,X3) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7461_tan__diff,axiom,
    ! [X3: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X3) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X3),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X3),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X3)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X3)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7462_tan__diff,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(cos_real,X3) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(X3),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(X3),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7463_tan__add,axiom,
    ! [X3: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X3) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X3),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X3),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X3)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X3)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7464_tan__add,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(cos_real,X3) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(plus_plus_real(X3),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7465_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7466_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_7467_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_7468_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7469_exp__of__nat2__mult,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(X3),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X3)),Nb) ).

% exp_of_nat2_mult
tff(fact_7470_exp__of__nat2__mult,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(times_times_real(X3),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(power_power_real(aa_real_real(exp_real,X3)),Nb) ).

% exp_of_nat2_mult
tff(fact_7471_exp__of__nat__mult,axiom,
    ! [Nb: nat,X3: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),X3)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X3)),Nb) ).

% exp_of_nat_mult
tff(fact_7472_exp__of__nat__mult,axiom,
    ! [Nb: nat,X3: real] : aa_real_real(exp_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X3)) = aa_nat_real(power_power_real(aa_real_real(exp_real,X3)),Nb) ).

% exp_of_nat_mult
tff(fact_7473_cos__one__2pi,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) = one_one_real )
    <=> ( ? [X4: nat] : X3 = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X4: nat] : X3 = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7474_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,K)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_7475_cos__add,axiom,
    ! [X3: real,Y: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),Y)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_7476_cos__diff,axiom,
    ! [X3: real,Y: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(X3),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_7477_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7478_mod__mult2__eq_H,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))) ).

% mod_mult2_eq'
tff(fact_7479_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_7480_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_7481_tan__half,axiom,
    ! [X3: complex] : aa_complex_complex(tan_complex,X3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3))),one_one_complex)) ).

% tan_half
tff(fact_7482_tan__half,axiom,
    ! [X3: real] : aa_real_real(tan_real,X3) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3))),aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3))),one_one_real)) ).

% tan_half
tff(fact_7483_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_7484_less__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Mb)) ) ) ).

% less_log_of_power
tff(fact_7485_log__of__power__eq,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(power_power_real(B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_7486_sin__expansion__lemma,axiom,
    ! [X3: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7487_sin__double,axiom,
    ! [X3: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,X3))),aa_complex_complex(cos_complex,X3)) ).

% sin_double
tff(fact_7488_sin__double,axiom,
    ! [X3: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,X3))),aa_real_real(cos_real,X3)) ).

% sin_double
tff(fact_7489_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_7490_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_7491_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7492_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7493_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7494_cos__zero__lemma,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( ( aa_real_real(cos_real,X3) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( X3 = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7495_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_7496_cos__is__zero,axiom,
    ? [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X5)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X5) = zero_zero_real )
      & ! [Y5: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y5) = zero_zero_real ) )
         => ( Y5 = X5 ) ) ) ).

% cos_is_zero
tff(fact_7497_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_7498_cos__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) = zero_zero_real )
    <=> ( ? [N3: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( X3 = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N3: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( X3 = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7499_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_7500_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != 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,Mb))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_7501_cos__expansion__lemma,axiom,
    ! [X3: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7502_tan__double,axiom,
    ! [X3: complex] :
      ( ( aa_complex_complex(cos_complex,X3) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X3))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7503_tan__double,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,X3))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7504_exp__divide__power__eq,axiom,
    ! [Nb: nat,X3: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X3),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,X3) ) ) ).

% exp_divide_power_eq
tff(fact_7505_exp__divide__power__eq,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(exp_real,aa_real_real(divide_divide_real(X3),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,X3) ) ) ).

% exp_divide_power_eq
tff(fact_7506_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Mb)) ) ) ).

% le_log_of_power
tff(fact_7507_log__base__pow,axiom,
    ! [A: real,Nb: nat,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(log(aa_nat_real(power_power_real(A),Nb)),X3) = aa_real_real(divide_divide_real(aa_real_real(log(A),X3)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_7508_ln__realpow,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(ln_ln_real,aa_nat_real(power_power_real(X3),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,X3)) ) ) ).

% ln_realpow
tff(fact_7509_log__nat__power,axiom,
    ! [X3: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(log(B),aa_nat_real(power_power_real(X3),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),X3)) ) ) ).

% log_nat_power
tff(fact_7510_cos__tan,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,X3) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7511_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_7512_cos__one__2pi__int,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) = one_one_real )
    <=> ? [X4: int] : X3 = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7513_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_7514_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7515_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).

% log2_of_power_eq
tff(fact_7516_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(power_power_real(B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_7517_Bernoulli__inequality,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X3))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(one_one_real),X3)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_7518_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7519_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7520_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7521_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7522_cos__squared__eq,axiom,
    ! [X3: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7523_cos__squared__eq,axiom,
    ! [X3: real] : aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(sin_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7524_sin__squared__eq,axiom,
    ! [X3: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7525_sin__squared__eq,axiom,
    ! [X3: real] : aa_nat_real(power_power_real(aa_real_real(sin_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7526_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X5)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X5)) ) ) ).

% lemma_tan_total
tff(fact_7527_tan__gt__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(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,X3)) ) ) ).

% tan_gt_zero
tff(fact_7528_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X5)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X5) = Y ) ) ).

% lemma_tan_total1
tff(fact_7529_tan__mono__lt__eq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_7530_tan__monotone_H,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X3)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_7531_tan__monotone,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X3)) ) ) ) ).

% tan_monotone
tff(fact_7532_tan__total,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X5)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X5) = Y )
      & ! [Y5: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y5) = Y ) )
         => ( Y5 = X5 ) ) ) ).

% tan_total
tff(fact_7533_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7534_cos__double__less__one,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_7535_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_7536_cos__gt__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(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,X3)) ) ) ).

% cos_gt_zero
tff(fact_7537_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_7538_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_7539_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(power_power_real(B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_7540_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7541_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_7542_tan__cot_H,axiom,
    ! [X3: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)) = aa_real_real(cot_real,X3) ).

% tan_cot'
tff(fact_7543_cos__treble__cos,axiom,
    ! [X3: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X3)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X3))) ).

% cos_treble_cos
tff(fact_7544_cos__treble__cos,axiom,
    ! [X3: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),X3)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,X3))) ).

% cos_treble_cos
tff(fact_7545_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7546_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7547_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7548_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7549_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7550_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7551_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7552_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7553_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7554_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7555_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7556_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7557_cos__double,axiom,
    ! [X3: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X3)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7558_cos__double,axiom,
    ! [X3: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7559_tan__pos__pi2__le,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X3)) ) ) ).

% tan_pos_pi2_le
tff(fact_7560_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X5)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X5) = Y ) ) ) ).

% tan_total_pos
tff(fact_7561_tan__less__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X3)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_7562_tan__mono__le__eq,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_7563_tan__mono__le,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_7564_tan__bound__pi2,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X3))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_7565_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_7566_cos__gt__zero__pi,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X3)) ) ) ).

% cos_gt_zero_pi
tff(fact_7567_cos__ge__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X3)) ) ) ).

% cos_ge_zero
tff(fact_7568_arctan__unique,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,X3) = Y )
         => ( aa_real_real(arctan,Y) = X3 ) ) ) ) ).

% arctan_unique
tff(fact_7569_arctan__tan,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X3)) = X3 ) ) ) ).

% arctan_tan
tff(fact_7570_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_7571_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_7572_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_7573_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7574_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7575_tan__total__pi4,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),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(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(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) = X3 ) ) ) ).

% tan_total_pi4
tff(fact_7576_cos__zero__iff__int,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4)
          & ( X3 = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7577_cos__arctan,axiom,
    ! [X3: real] : aa_real_real(cos_real,aa_real_real(arctan,X3)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7578_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7579_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_7580_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X3))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(one_one_real),X3)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_7581_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_7582_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(X3),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,X3)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7583_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X3),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X3))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7584_sincos__total__pi,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [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)
            & ( X3 = aa_real_real(cos_real,T4) )
            & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_7585_sin__cos__sqrt,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X3))
     => ( aa_real_real(sin_real,X3) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7586_cos__arcsin,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X3)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_7587_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_7588_sin__zero__lemma,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( ( aa_real_real(sin_real,X3) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( X3 = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7589_sin__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sin_real,X3) = zero_zero_real )
    <=> ( ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( X3 = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( X3 = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7590_sin__arccos,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,X3)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_7591_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_hq,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7592_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,aa_Pro3608043709957902237at_rat(produc6207742614233964070at_rat(aTP_Lamm_hr),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7593_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_hs),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7594_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_ht),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7595_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_hu),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7596_linear__plus__1__le__power,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X3)),one_one_real)),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(X3),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_7597_monoseq__arctan__series,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => topolo6980174941875973593q_real(aTP_Lamm_hv(X3)) ) ).

% monoseq_arctan_series
tff(fact_7598_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(power_power_real(Z),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7599_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( 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))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7600_ln__series,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,X3) = suminf_real(aTP_Lamm_hw(X3)) ) ) ) ).

% ln_series
tff(fact_7601_arctan__series,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => ( aa_real_real(arctan,X3) = suminf_real(aTP_Lamm_hx(X3)) ) ) ).

% arctan_series
tff(fact_7602_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V2: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V2) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V2) ) ) ).

% int_eq_iff_numeral
tff(fact_7603_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_7604_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_hy(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7605_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_hz(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7606_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_7607_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_7608_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_7609_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_7610_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_7611_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_7612_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_7613_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_7614_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_7615_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_7616_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_7617_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_7618_monoseq__realpow,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => topolo6980174941875973593q_real(power_power_real(X3)) ) ) ).

% monoseq_realpow
tff(fact_7619_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_7620_negD,axiom,
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int)
     => ? [N: nat] : X3 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_7621_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7622_zdiff__int__split,axiom,
    ! [P: fun_int_o,X3: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7623_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(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_7624_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(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_7625_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(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(plus_plus_real(one_one_real),aa_real_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_7626_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(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(plus_plus_real(one_one_real),aa_real_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_7627_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ia) ).

% pi_series
tff(fact_7628_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7629_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7630_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_7631_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_7632_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7633_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7634_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7635_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7636_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_real(power_power_real(C)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7637_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7638_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7639_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7640_norm__power,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(power_power_real(X3),Nb)) = aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,X3)),Nb) ).

% norm_power
tff(fact_7641_norm__power,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X3),Nb)) = aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,X3)),Nb) ).

% norm_power
tff(fact_7642_norm__uminus__minus,axiom,
    ! [X3: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X3)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X3),Y)) ).

% norm_uminus_minus
tff(fact_7643_norm__uminus__minus,axiom,
    ! [X3: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X3)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X3),Y)) ).

% norm_uminus_minus
tff(fact_7644_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(power_power_real(W),Nb) = aa_nat_real(power_power_real(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7645_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7646_norm__triangle__lt,axiom,
    ! [X3: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X3)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X3),Y))),E) ) ).

% norm_triangle_lt
tff(fact_7647_norm__triangle__lt,axiom,
    ! [X3: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X3)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X3),Y))),E) ) ).

% norm_triangle_lt
tff(fact_7648_norm__add__less,axiom,
    ! [X3: real,R2: real,Y: real,Sb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),Sb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X3),Y))),aa_real_real(plus_plus_real(R2),Sb)) ) ) ).

% norm_add_less
tff(fact_7649_norm__add__less,axiom,
    ! [X3: complex,R2: real,Y: complex,Sb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),Sb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X3),Y))),aa_real_real(plus_plus_real(R2),Sb)) ) ) ).

% norm_add_less
tff(fact_7650_norm__power__ineq,axiom,
    ! [X3: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(power_power_real(X3),Nb))),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,X3)),Nb)) ).

% norm_power_ineq
tff(fact_7651_norm__power__ineq,axiom,
    ! [X3: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X3),Nb))),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,X3)),Nb)) ).

% norm_power_ineq
tff(fact_7652_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,Sb: 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,B)),Sb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))),aa_real_real(plus_plus_real(R2),Sb)) ) ) ).

% norm_triangle_mono
tff(fact_7653_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,Sb: 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,B)),Sb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(plus_plus_real(R2),Sb)) ) ) ).

% norm_triangle_mono
tff(fact_7654_norm__triangle__ineq,axiom,
    ! [X3: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X3),Y))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X3)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_7655_norm__triangle__ineq,axiom,
    ! [X3: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X3),Y))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X3)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_7656_norm__triangle__le,axiom,
    ! [X3: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X3)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X3),Y))),E) ) ).

% norm_triangle_le
tff(fact_7657_norm__triangle__le,axiom,
    ! [X3: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X3)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X3),Y))),E) ) ).

% norm_triangle_le
tff(fact_7658_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_7659_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: 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),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_7660_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))) ).

% norm_diff_ineq
tff(fact_7661_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_7662_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(power_power_real(W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7663_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7664_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7665_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7666_square__norm__one,axiom,
    ! [X3: real] :
      ( ( aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X3) = one_one_real ) ) ).

% square_norm_one
tff(fact_7667_square__norm__one,axiom,
    ! [X3: complex] :
      ( ( aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X3) = one_one_real ) ) ).

% square_norm_one
tff(fact_7668_norm__power__diff,axiom,
    ! [Z: real,W: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Z),Mb)),aa_nat_real(power_power_real(W),Mb)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_7669_norm__power__diff,axiom,
    ! [Z: complex,W: complex,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),Mb)),aa_nat_complex(power_power_complex(W),Mb)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_7670_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_7671_summable__arctan__series,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => summable_real(aTP_Lamm_hx(X3)) ) ).

% summable_arctan_series
tff(fact_7672_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_7673_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_7674_sum__gp,axiom,
    ! [X3: complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(X3 = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X3),Mb)),aa_nat_complex(power_power_complex(X3),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),X3))) ) ).

% sum_gp
tff(fact_7675_sum__gp,axiom,
    ! [X3: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_rat,
        $ite(X3 = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X3),Mb)),aa_nat_rat(power_power_rat(X3),aa_nat_nat(suc,Nb)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3))) ) ).

% sum_gp
tff(fact_7676_sum__gp,axiom,
    ! [X3: real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(X3 = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),Mb)),aa_nat_real(power_power_real(X3),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),X3))) ) ).

% sum_gp
tff(fact_7677_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7678_sum_Oempty,axiom,
    ! [G: fun_o_real] : groups8691415230153176458o_real(G,bot_bot_set_o) = zero_zero_real ).

% sum.empty
tff(fact_7679_sum_Oempty,axiom,
    ! [G: fun_o_rat] : groups7872700643590313910_o_rat(G,bot_bot_set_o) = zero_zero_rat ).

% sum.empty
tff(fact_7680_sum_Oempty,axiom,
    ! [G: fun_o_nat] : groups8507830703676809646_o_nat(G,bot_bot_set_o) = zero_zero_nat ).

% sum.empty
tff(fact_7681_sum_Oempty,axiom,
    ! [G: fun_o_int] : groups8505340233167759370_o_int(G,bot_bot_set_o) = zero_zero_int ).

% sum.empty
tff(fact_7682_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_7683_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_7684_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_7685_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_7686_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_7687_sum_Oempty,axiom,
    ! [G: fun_nat_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),bot_bot_set_nat) = zero_zero_nat ).

% sum.empty
tff(fact_7688_ceiling__numeral,axiom,
    ! [V2: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V2)) = aa_num_int(numeral_numeral_int,V2) ).

% ceiling_numeral
tff(fact_7689_ceiling__numeral,axiom,
    ! [V2: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V2)) = aa_num_int(numeral_numeral_int,V2) ).

% ceiling_numeral
tff(fact_7690_sum__abs,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(F,A4))),groups6591440286371151544t_real(aTP_Lamm_ic(F),A4)) ).

% sum_abs
tff(fact_7691_sum__abs,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A4))),groups4538972089207619220nt_int(aTP_Lamm_id(F),A4)) ).

% sum_abs
tff(fact_7692_sum_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(X3),A4)
       => ( groups8097168146408367636l_real(G,insert_real(X3,A4)) = aa_real_real(plus_plus_real(aa_real_real(G,X3)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7693_sum_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X3)),A4)
       => ( groups8691415230153176458o_real(G,insert_o((X3),A4)) = aa_real_real(plus_plus_real(aa_o_real(G,(X3))),groups8691415230153176458o_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7694_sum_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(X3),A4)
       => ( groups8778361861064173332t_real(G,insert_int(X3,A4)) = aa_real_real(plus_plus_real(aa_int_real(G,X3)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7695_sum_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(X3),A4)
       => ( groups5808333547571424918x_real(G,insert_complex(X3,A4)) = aa_real_real(plus_plus_real(aa_complex_real(G,X3)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7696_sum_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(X3),A4)
       => ( groups1300246762558778688al_rat(G,insert_real(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X3)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7697_sum_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X3)),A4)
       => ( groups7872700643590313910_o_rat(G,insert_o((X3),A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X3))),groups7872700643590313910_o_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7698_sum_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(X3),A4)
       => ( groups3906332499630173760nt_rat(G,insert_int(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X3)),groups3906332499630173760nt_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7699_sum_Oinsert,axiom,
    ! [A4: set_nat,X3: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(X3),A4)
       => ( groups2906978787729119204at_rat(G,insert_nat(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X3)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7700_sum_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(X3),A4)
       => ( groups5058264527183730370ex_rat(G,insert_complex(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X3)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7701_sum_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(X3),A4)
       => ( groups1935376822645274424al_nat(G,insert_real(X3,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X3)),groups1935376822645274424al_nat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_7702_ceiling__add__of__int,axiom,
    ! [X3: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),Z) ).

% ceiling_add_of_int
tff(fact_7703_ceiling__add__of__int,axiom,
    ! [X3: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X3),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X3)),Z) ).

% ceiling_add_of_int
tff(fact_7704_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_ic(F),A4)) ).

% sum_abs_ge_zero
tff(fact_7705_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_id(F),A4)) ).

% sum_abs_ge_zero
tff(fact_7706_ceiling__le__zero,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X3)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_7707_ceiling__le__zero,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_7708_ceiling__le__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_num_real(numeral_numeral_real,V2)) ) ).

% ceiling_le_numeral
tff(fact_7709_ceiling__le__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_num_rat(numeral_numeral_rat,V2)) ) ).

% ceiling_le_numeral
tff(fact_7710_ceiling__less__one,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X3)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_7711_ceiling__less__one,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_7712_numeral__less__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V2)),aa_real_int(archim7802044766580827645g_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V2)),X3) ) ).

% numeral_less_ceiling
tff(fact_7713_numeral__less__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V2)),aa_rat_int(archim2889992004027027881ng_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V2)),X3) ) ).

% numeral_less_ceiling
tff(fact_7714_ceiling__le__one,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X3)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real) ) ).

% ceiling_le_one
tff(fact_7715_ceiling__le__one,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),one_one_rat) ) ).

% ceiling_le_one
tff(fact_7716_ceiling__add__numeral,axiom,
    ! [X3: real,V2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X3),aa_num_real(numeral_numeral_real,V2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_num_int(numeral_numeral_int,V2)) ).

% ceiling_add_numeral
tff(fact_7717_ceiling__add__numeral,axiom,
    ! [X3: rat,V2: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_num_rat(numeral_numeral_rat,V2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_num_int(numeral_numeral_int,V2)) ).

% ceiling_add_numeral
tff(fact_7718_ceiling__neg__numeral,axiom,
    ! [V2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)) ).

% ceiling_neg_numeral
tff(fact_7719_ceiling__neg__numeral,axiom,
    ! [V2: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)) ).

% ceiling_neg_numeral
tff(fact_7720_ceiling__add__one,axiom,
    ! [X3: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),one_one_int) ).

% ceiling_add_one
tff(fact_7721_ceiling__add__one,axiom,
    ! [X3: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X3),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X3)),one_one_int) ).

% ceiling_add_one
tff(fact_7722_ceiling__diff__numeral,axiom,
    ! [X3: real,V2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(X3),aa_num_real(numeral_numeral_real,V2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_num_int(numeral_numeral_int,V2)) ).

% ceiling_diff_numeral
tff(fact_7723_ceiling__diff__numeral,axiom,
    ! [X3: rat,V2: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_num_rat(numeral_numeral_rat,V2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_num_int(numeral_numeral_int,V2)) ).

% ceiling_diff_numeral
tff(fact_7724_ceiling__numeral__power,axiom,
    ! [X3: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ).

% ceiling_numeral_power
tff(fact_7725_ceiling__numeral__power,axiom,
    ! [X3: num,Nb: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X3)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ).

% ceiling_numeral_power
tff(fact_7726_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(power_power_real(C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7727_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7728_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7729_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7730_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7731_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7732_ceiling__less__zero,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X3)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_7733_ceiling__less__zero,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_7734_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A4: set_nat] :
      groups2073611262835488442omplex(aTP_Lamm_hy(C),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_7735_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A4: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ie(C),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_7736_sum__zero__power,axiom,
    ! [C: fun_nat_real,A4: set_nat] :
      groups6591440286371151544t_real(aTP_Lamm_hz(C),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_7737_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_7738_ceiling__less__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V2)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_7739_ceiling__less__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V2)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_7740_numeral__le__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V2)),aa_real_int(archim7802044766580827645g_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V2)),one_one_real)),X3) ) ).

% numeral_le_ceiling
tff(fact_7741_numeral__le__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V2)),aa_rat_int(archim2889992004027027881ng_rat,X3))
    <=> 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,V2)),one_one_rat)),X3) ) ).

% numeral_le_ceiling
tff(fact_7742_ceiling__le__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) ) ).

% ceiling_le_neg_numeral
tff(fact_7743_ceiling__le__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))) ) ).

% ceiling_le_neg_numeral
tff(fact_7744_neg__numeral__less__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_real_int(archim7802044766580827645g_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),X3) ) ).

% neg_numeral_less_ceiling
tff(fact_7745_neg__numeral__less__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_rat_int(archim2889992004027027881ng_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),X3) ) ).

% neg_numeral_less_ceiling
tff(fact_7746_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A4: set_nat] :
      groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_if(C),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_7747_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A4: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ig(C),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_7748_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A4: set_nat] :
      groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ih(C),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_real_real(divide_divide_real(aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_7749_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_7750_ceiling__less__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_7751_ceiling__less__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_7752_neg__numeral__le__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_real_int(archim7802044766580827645g_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),one_one_real)),X3) ) ).

% neg_numeral_le_ceiling
tff(fact_7753_neg__numeral__le__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_rat_int(archim2889992004027027881ng_rat,X3))
    <=> 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,V2))),one_one_rat)),X3) ) ).

% neg_numeral_le_ceiling
tff(fact_7754_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7755_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7756_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7757_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7758_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(F),G)) ) ) ).

% summable_add
tff(fact_7759_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(F),G)) ) ) ).

% summable_add
tff(fact_7760_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ik(F),G)) ) ) ).

% summable_add
tff(fact_7761_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_il(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7762_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2)) ) ).

% summable_ignore_initial_segment
tff(fact_7763_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).

% sum_mono
tff(fact_7764_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_7765_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,K5)),groups7872700643590313910_o_rat(G,K5)) ) ).

% sum_mono
tff(fact_7766_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_7767_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_7768_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).

% sum_mono
tff(fact_7769_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_7770_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I3))),aa_o_nat(G,(I3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,K5)),groups8507830703676809646_o_nat(G,K5)) ) ).

% sum_mono
tff(fact_7771_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_7772_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).

% sum_mono
tff(fact_7773_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(G),H)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A4)) ).

% sum.distrib
tff(fact_7774_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A4: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_im(G),H),A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A4)),groups7754918857620584856omplex(H,A4)) ).

% sum.distrib
tff(fact_7775_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A4: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(G),H),A4) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,A4)),groups6591440286371151544t_real(H,A4)) ).

% sum.distrib
tff(fact_7776_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_in(G),H),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A4)),groups4538972089207619220nt_int(H,A4)) ).

% sum.distrib
tff(fact_7777_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I5: set_nat] :
      ( summable_int(F)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,I5)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7778_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I5: set_nat] :
      ( summable_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7779_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I5: set_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,I5)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7780_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X5)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7781_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X5)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7782_sum__nonpos,axiom,
    ! [A4: set_o,F: fun_o_real] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(X5))),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8691415230153176458o_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7783_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X5)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7784_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X5)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7785_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X5)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7786_sum__nonpos,axiom,
    ! [A4: set_o,F: fun_o_rat] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(X5))),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7787_sum__nonpos,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7788_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X5)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7789_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X5)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7790_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A4)) ) ).

% sum_nonneg
tff(fact_7791_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A4)) ) ).

% sum_nonneg
tff(fact_7792_sum__nonneg,axiom,
    ! [A4: set_o,F: fun_o_real] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X5))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8691415230153176458o_real(F,A4)) ) ).

% sum_nonneg
tff(fact_7793_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A4)) ) ).

% sum_nonneg
tff(fact_7794_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_7795_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_7796_sum__nonneg,axiom,
    ! [A4: set_o,F: fun_o_rat] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X5))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_7797_sum__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_7798_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_7799_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X5)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_7800_powser__insidea,axiom,
    ! [F: fun_nat_real,X3: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),X3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X3))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ip(F),Z)) ) ) ).

% powser_insidea
tff(fact_7801_powser__insidea,axiom,
    ! [F: fun_nat_complex,X3: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),X3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X3))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_ir(F),Z)) ) ) ).

% powser_insidea
tff(fact_7802_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7803_sum__mono__inv,axiom,
    ! [F: fun_o_rat,I5: set_o,G: fun_o_rat,I: $o] :
      ( ( groups7872700643590313910_o_rat(F,I5) = groups7872700643590313910_o_rat(G,I5) )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) )
       => ( aa_set_o_o(member_o((I)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_rat(F,(I)) = aa_o_rat(G,(I)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7804_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7805_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7806_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7807_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7808_sum__mono__inv,axiom,
    ! [F: fun_o_nat,I5: set_o,G: fun_o_nat,I: $o] :
      ( ( groups8507830703676809646_o_nat(F,I5) = groups8507830703676809646_o_nat(G,I5) )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I3))),aa_o_nat(G,(I3))) )
       => ( aa_set_o_o(member_o((I)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_nat(F,(I)) = aa_o_nat(G,(I)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7809_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7810_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7811_sum__mono__inv,axiom,
    ! [F: fun_real_int,I5: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I5) = groups1932886352136224148al_int(G,I5) )
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7812_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( summable_real(F)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_7813_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_7814_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( summable_int(F)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_7815_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X5)),A4)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X5)) = aa_nat_nat(G,aa_nat_nat(suc,X5)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_7816_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X5)),A4)
           => ( aa_nat_real(F,aa_nat_nat(suc,X5)) = aa_nat_real(G,aa_nat_nat(suc,X5)) ) )
       => ( groups6591440286371151544t_real(F,A4) = groups6591440286371151544t_real(G,A4) ) ) ) ).

% sum_cong_Suc
tff(fact_7817_ceiling__mono,axiom,
    ! [Y: real,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X3)) ) ).

% ceiling_mono
tff(fact_7818_ceiling__mono,axiom,
    ! [Y: rat,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X3)) ) ).

% ceiling_mono
tff(fact_7819_le__of__int__ceiling,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X3))) ).

% le_of_int_ceiling
tff(fact_7820_le__of__int__ceiling,axiom,
    ! [X3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X3))) ).

% le_of_int_ceiling
tff(fact_7821_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_7822_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7823_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7824_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(plus_plus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(F),G)) ) ) ) ).

% suminf_add
tff(fact_7825_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(F),G)) ) ) ) ).

% suminf_add
tff(fact_7826_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_ik(F),G)) ) ) ) ).

% suminf_add
tff(fact_7827_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7828_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups6591440286371151544t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7829_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N7)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7830_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N7)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7831_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_it(G),K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7832_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K2: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ib(G),K2),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7833_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_nonneg
tff(fact_7834_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_nonneg
tff(fact_7835_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_nonneg
tff(fact_7836_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7837_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7838_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7839_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5)) )
       => ( ( groups8097168146408367636l_real(F,A4) = zero_zero_real )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A4)
             => ( aa_real_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7840_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X5))) )
       => ( ( groups8691415230153176458o_real(F,A4) = zero_zero_real )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => ( aa_o_real(F,(X4)) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7841_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5)) )
       => ( ( groups8778361861064173332t_real(F,A4) = zero_zero_real )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7842_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5)) )
       => ( ( groups5808333547571424918x_real(F,A4) = zero_zero_real )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A4)
             => ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7843_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5)) )
       => ( ( groups1300246762558778688al_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A4)
             => ( aa_real_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7844_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X5))) )
       => ( ( groups7872700643590313910_o_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => ( aa_o_rat(F,(X4)) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7845_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5)) )
       => ( ( groups3906332499630173760nt_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7846_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5)) )
       => ( ( groups2906978787729119204at_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7847_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5)) )
       => ( ( groups5058264527183730370ex_rat(F,A4) = zero_zero_rat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A4)
             => ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7848_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X5)) )
       => ( ( groups1935376822645274424al_nat(F,A4) = zero_zero_nat )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A4)
             => ( aa_real_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7849_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X5)) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),Sb)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X5 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X5)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,Sb)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7850_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X5)) )
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),Sb)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X5 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,Sb)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7851_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X5)) )
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),Sb)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_nat(I,Xa) = X5 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,Sb)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7852_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_nat,G: fun_nat_rat,I: fun_nat_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X5)) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),Sb)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_complex(I,Xa) = X5 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X5)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,Sb)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7853_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_complex,G: fun_complex_rat,I: fun_complex_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X5)) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),Sb)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X5 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X5)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,Sb)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7854_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_complex,G: fun_complex_nat,I: fun_complex_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),Ta)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(G,X5)) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),Sb)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X5 )
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X5)),aa_complex_nat(G,Xa)) ) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,Sb)),groups5693394587270226106ex_nat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7855_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_nat,G: fun_nat_int,I: fun_nat_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(G,X5)) )
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),Sb)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X5 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X5)),aa_nat_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,Sb)),groups3539618377306564664at_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7856_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_complex,G: fun_complex_int,I: fun_complex_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(G,X5)) )
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),Sb)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_nat(I,Xa) = X5 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X5)),aa_complex_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,Sb)),groups5690904116761175830ex_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7857_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_nat,G: fun_nat_int,I: fun_nat_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(G,X5)) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),Sb)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_complex(I,Xa) = X5 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X5)),aa_nat_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,Sb)),groups3539618377306564664at_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7858_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_complex,G: fun_complex_int,I: fun_complex_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(G,X5)) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),Sb)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X5 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X5)),aa_complex_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,Sb)),groups5690904116761175830ex_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7859_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X5)),aa_complex_real(G,X5)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7860_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(G,X5)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7861_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X5)),aa_complex_rat(G,X5)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7862_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X5)),aa_complex_nat(G,X5)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7863_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X5)),aa_nat_int(G,X5)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7864_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X5)),aa_complex_int(G,X5)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7865_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X5)),aa_nat_nat(G,X5)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7866_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X5)),aa_nat_real(G,X5)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7867_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X5)),aa_int_int(G,X5)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7868_sum__strict__mono__ex1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real,G: fun_Pr672294278902479040t_real] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [X5: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Pro6168199777849092593t_real(F,X5)),aa_Pro6168199777849092593t_real(G,X5)) )
       => ( ? [X: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Pro6168199777849092593t_real(F,X)),aa_Pro6168199777849092593t_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups4567486121110086003t_real(F,A4)),groups4567486121110086003t_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7869_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X5)),aa_complex_real(G,X5)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7870_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X5)),aa_nat_rat(G,X5)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7871_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X5)),aa_complex_rat(G,X5)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7872_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X5)),aa_complex_nat(G,X5)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7873_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X5)),aa_nat_int(G,X5)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7874_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_complex_int(H,X5)),aa_complex_int(G,X5)) )
           => aa_int_o(aa_int_fun_int_o(R,groups5690904116761175830ex_int(H,S2)),groups5690904116761175830ex_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7875_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_nat,H: fun_nat_nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(H,X5)),aa_nat_nat(G,X5)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_set_nat_nat(groups3542108847815614940at_nat(H),S2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),S2)) ) ) ) ) ).

% sum.related
tff(fact_7876_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(H,X5)),aa_complex_complex(G,X5)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups7754918857620584856omplex(H,S2)),groups7754918857620584856omplex(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7877_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X5)),aa_nat_real(G,X5)) )
           => aa_real_o(aa_real_fun_real_o(R,groups6591440286371151544t_real(H,S2)),groups6591440286371151544t_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7878_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_int,H: fun_int_int,G: fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_int_int(H,X5)),aa_int_int(G,X5)) )
           => aa_int_o(aa_int_fun_int_o(R,groups4538972089207619220nt_int(H,S2)),groups4538972089207619220nt_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7879_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X5)),aa_real_real(G,X5)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7880_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X5)),aa_complex_real(G,X5)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7881_sum__strict__mono,axiom,
    ! [A4: set_o,F: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(X5))),aa_o_real(G,(X5))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8691415230153176458o_real(F,A4)),groups8691415230153176458o_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7882_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X5)),aa_int_real(G,X5)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7883_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7884_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X5)),aa_complex_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7885_sum__strict__mono,axiom,
    ! [A4: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(X5))),aa_o_rat(G,(X5))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A4)),groups7872700643590313910_o_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7886_sum__strict__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7887_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(G,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7888_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(G,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_7889_sum_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_real,X3: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups8097168146408367636l_real(G,insert_real(X3,A4)) = $ite(aa_set_real_o(member_real(X3),A4),groups8097168146408367636l_real(G,A4),aa_real_real(plus_plus_real(aa_real_real(G,X3)),groups8097168146408367636l_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7890_sum_Oinsert__if,axiom,
    ! [A4: set_o,G: fun_o_real,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8691415230153176458o_real(G,insert_o((X3),A4)) = $ite(aa_set_o_o(member_o((X3)),A4),groups8691415230153176458o_real(G,A4),aa_real_real(plus_plus_real(aa_o_real(G,(X3))),groups8691415230153176458o_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7891_sum_Oinsert__if,axiom,
    ! [A4: set_int,G: fun_int_real,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,insert_int(X3,A4)) = $ite(aa_set_int_o(member_int(X3),A4),groups8778361861064173332t_real(G,A4),aa_real_real(plus_plus_real(aa_int_real(G,X3)),groups8778361861064173332t_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7892_sum_Oinsert__if,axiom,
    ! [A4: set_complex,G: fun_complex_real,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,insert_complex(X3,A4)) = $ite(aa_set_complex_o(member_complex(X3),A4),groups5808333547571424918x_real(G,A4),aa_real_real(plus_plus_real(aa_complex_real(G,X3)),groups5808333547571424918x_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7893_sum_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_rat,X3: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups1300246762558778688al_rat(G,insert_real(X3,A4)) = $ite(aa_set_real_o(member_real(X3),A4),groups1300246762558778688al_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X3)),groups1300246762558778688al_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7894_sum_Oinsert__if,axiom,
    ! [A4: set_o,G: fun_o_rat,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(G,insert_o((X3),A4)) = $ite(aa_set_o_o(member_o((X3)),A4),groups7872700643590313910_o_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X3))),groups7872700643590313910_o_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7895_sum_Oinsert__if,axiom,
    ! [A4: set_int,G: fun_int_rat,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,insert_int(X3,A4)) = $ite(aa_set_int_o(member_int(X3),A4),groups3906332499630173760nt_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X3)),groups3906332499630173760nt_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7896_sum_Oinsert__if,axiom,
    ! [A4: set_nat,G: fun_nat_rat,X3: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(G,insert_nat(X3,A4)) = $ite(aa_set_nat_o(member_nat(X3),A4),groups2906978787729119204at_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X3)),groups2906978787729119204at_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7897_sum_Oinsert__if,axiom,
    ! [A4: set_complex,G: fun_complex_rat,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,insert_complex(X3,A4)) = $ite(aa_set_complex_o(member_complex(X3),A4),groups5058264527183730370ex_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X3)),groups5058264527183730370ex_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7898_sum_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_nat,X3: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups1935376822645274424al_nat(G,insert_real(X3,A4)) = $ite(aa_set_real_o(member_real(X3),A4),groups1935376822645274424al_nat(G,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X3)),groups1935376822645274424al_nat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_7899_ceiling__le,axiom,
    ! [X3: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),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,X3)),A) ) ).

% ceiling_le
tff(fact_7900_ceiling__le,axiom,
    ! [X3: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),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,X3)),A) ) ).

% ceiling_le
tff(fact_7901_ceiling__le__iff,axiom,
    ! [X3: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X3)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_7902_ceiling__le__iff,axiom,
    ! [X3: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_7903_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_hy(F)) ).

% summable_zero_power'
tff(fact_7904_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_hz(F)) ).

% summable_zero_power'
tff(fact_7905_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_iu(F)) ).

% summable_zero_power'
tff(fact_7906_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_hy(F)) ).

% summable_0_powser
tff(fact_7907_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_hz(F)) ).

% summable_0_powser
tff(fact_7908_ceiling__add__le,axiom,
    ! [X3: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_7909_ceiling__add__le,axiom,
    ! [X3: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X3),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X3)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_7910_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7911_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_iw(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7912_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7913_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_iw(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7914_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ix(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7915_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Mb: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_iy(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7916_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_real,F: fun_real_real,B4: real,I: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,Sb) = B4 )
         => ( aa_set_real_o(member_real(I),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7917_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_o,F: fun_o_real,B4: real,I: $o] :
      ( aa_set_o_o(finite_finite_o,Sb)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I3))) )
       => ( ( groups8691415230153176458o_real(F,Sb) = B4 )
         => ( aa_set_o_o(member_o((I)),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I))),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7918_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_int,F: fun_int_real,B4: real,I: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,Sb) = B4 )
         => ( aa_set_int_o(member_int(I),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7919_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,Sb) = B4 )
         => ( aa_set_complex_o(member_complex(I),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7920_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_real,F: fun_real_rat,B4: rat,I: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,Sb) = B4 )
         => ( aa_set_real_o(member_real(I),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7921_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_o,F: fun_o_rat,B4: rat,I: $o] :
      ( aa_set_o_o(finite_finite_o,Sb)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
       => ( ( groups7872700643590313910_o_rat(F,Sb) = B4 )
         => ( aa_set_o_o(member_o((I)),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I))),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7922_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_int,F: fun_int_rat,B4: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,Sb) = B4 )
         => ( aa_set_int_o(member_int(I),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7923_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_nat,F: fun_nat_rat,B4: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,Sb) = B4 )
         => ( aa_set_nat_o(member_nat(I),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7924_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,Sb) = B4 )
         => ( aa_set_complex_o(member_complex(I),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7925_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_real,F: fun_real_nat,B4: nat,I: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
       => ( ( groups1935376822645274424al_nat(F,Sb) = B4 )
         => ( aa_set_real_o(member_real(I),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7926_sum__nonneg__0,axiom,
    ! [Sb: set_real,F: fun_real_real,I: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,Sb) = zero_zero_real )
         => ( aa_set_real_o(member_real(I),Sb)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7927_sum__nonneg__0,axiom,
    ! [Sb: set_o,F: fun_o_real,I: $o] :
      ( aa_set_o_o(finite_finite_o,Sb)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I3))) )
       => ( ( groups8691415230153176458o_real(F,Sb) = zero_zero_real )
         => ( aa_set_o_o(member_o((I)),Sb)
           => ( aa_o_real(F,(I)) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7928_sum__nonneg__0,axiom,
    ! [Sb: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,Sb) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),Sb)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7929_sum__nonneg__0,axiom,
    ! [Sb: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,Sb) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),Sb)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7930_sum__nonneg__0,axiom,
    ! [Sb: set_real,F: fun_real_rat,I: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_real_o(member_real(I),Sb)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7931_sum__nonneg__0,axiom,
    ! [Sb: set_o,F: fun_o_rat,I: $o] :
      ( aa_set_o_o(finite_finite_o,Sb)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
       => ( ( groups7872700643590313910_o_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_o_o(member_o((I)),Sb)
           => ( aa_o_rat(F,(I)) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7932_sum__nonneg__0,axiom,
    ! [Sb: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),Sb)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7933_sum__nonneg__0,axiom,
    ! [Sb: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),Sb)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7934_sum__nonneg__0,axiom,
    ! [Sb: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),Sb)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7935_sum__nonneg__0,axiom,
    ! [Sb: set_real,F: fun_real_nat,I: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
       => ( ( groups1935376822645274424al_nat(F,Sb) = zero_zero_nat )
         => ( aa_set_real_o(member_real(I),Sb)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7936_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_iz(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7937_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ic(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7938_sum__power__add,axiom,
    ! [X3: complex,Mb: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ja(X3),Mb),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X3),Mb)),groups2073611262835488442omplex(power_power_complex(X3),I5)) ).

% sum_power_add
tff(fact_7939_sum__power__add,axiom,
    ! [X3: rat,Mb: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_jb(X3),Mb),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),Mb)),groups2906978787729119204at_rat(power_power_rat(X3),I5)) ).

% sum_power_add
tff(fact_7940_sum__power__add,axiom,
    ! [X3: int,Mb: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_jc(X3),Mb),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X3),Mb)),groups3539618377306564664at_int(power_power_int(X3),I5)) ).

% sum_power_add
tff(fact_7941_sum__power__add,axiom,
    ! [X3: real,Mb: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jd(X3),Mb),I5) = aa_real_real(times_times_real(aa_nat_real(power_power_real(X3),Mb)),groups6591440286371151544t_real(power_power_real(X3),I5)) ).

% sum_power_add
tff(fact_7942_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_je(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7943_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_jf(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7944_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_7945_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_7946_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_7947_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7948_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7949_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7950_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7951_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I)),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(I)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I3))) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8691415230153176458o_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7952_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7953_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7954_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7955_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I)),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(I)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7956_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7957_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7958_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7959_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7960_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7961_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7962_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(I3))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8691415230153176458o_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7963_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7964_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7965_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7966_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(I3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7967_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7968_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7969_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7970_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_real(G,X5) = zero_zero_real ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_real(G,X5) = aa_real_real(H,X5) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7971_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S2: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_real(G,(X5)) = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_real(G,(X5)) = aa_o_real(H,(X5)) ) )
           => ( groups8691415230153176458o_real(G,T2) = groups8691415230153176458o_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7972_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_real(G,X5) = zero_zero_real ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_real(G,X5) = aa_int_real(H,X5) ) )
           => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7973_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X5) = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_real(G,X5) = aa_complex_real(H,X5) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7974_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(G,X5) = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_rat(G,X5) = aa_real_rat(H,X5) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7975_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S2: set_o,G: fun_o_rat,H: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_rat(G,(X5)) = zero_zero_rat ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_rat(G,(X5)) = aa_o_rat(H,(X5)) ) )
           => ( groups7872700643590313910_o_rat(G,T2) = groups7872700643590313910_o_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7976_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(G,X5) = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_rat(G,X5) = aa_int_rat(H,X5) ) )
           => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7977_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X5) = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_rat(G,X5) = aa_complex_rat(H,X5) ) )
           => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7978_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_nat(G,X5) = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_nat(G,X5) = aa_real_nat(H,X5) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7979_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_nat(G,(X5)) = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_nat(G,(X5)) = aa_o_nat(H,(X5)) ) )
           => ( groups8507830703676809646_o_nat(G,T2) = groups8507830703676809646_o_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7980_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_real(G,X5) = aa_real_real(H,X5) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7981_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S2: set_o,H: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_real(H,(X5)) = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_real(G,(X5)) = aa_o_real(H,(X5)) ) )
           => ( groups8691415230153176458o_real(G,S2) = groups8691415230153176458o_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7982_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_real(G,X5) = aa_int_real(H,X5) ) )
           => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7983_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_real(G,X5) = aa_complex_real(H,X5) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7984_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_rat(G,X5) = aa_real_rat(H,X5) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7985_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S2: set_o,H: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_rat(H,(X5)) = zero_zero_rat ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_rat(G,(X5)) = aa_o_rat(H,(X5)) ) )
           => ( groups7872700643590313910_o_rat(G,S2) = groups7872700643590313910_o_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7986_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_rat(G,X5) = aa_int_rat(H,X5) ) )
           => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7987_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_rat(G,X5) = aa_complex_rat(H,X5) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7988_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_nat(H,X5) = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_nat(G,X5) = aa_real_nat(H,X5) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7989_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S2: set_o,H: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_nat(H,(X5)) = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_nat(G,(X5)) = aa_o_nat(H,(X5)) ) )
           => ( groups8507830703676809646_o_nat(G,S2) = groups8507830703676809646_o_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7990_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X5) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7991_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X5) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7992_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X5) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7993_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X5) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7994_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_rat(G,X5) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7995_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_int(G,X5) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T2) = groups3539618377306564664at_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7996_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_nat(G,X5) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),T2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7997_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_complex(G,X5) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G,T2) = groups7754918857620584856omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7998_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_real(G,X5) = zero_zero_real ) )
         => ( groups6591440286371151544t_real(G,T2) = groups6591440286371151544t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7999_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_int(G,X5) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,T2) = groups4538972089207619220nt_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8000_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X5) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8001_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X5) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8002_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X5) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8003_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X5) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8004_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_rat(G,X5) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8005_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_int(G,X5) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S2) = groups3539618377306564664at_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8006_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_nat(G,X5) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8007_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_complex(G,X5) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G,S2) = groups7754918857620584856omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8008_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_real(G,X5) = zero_zero_real ) )
         => ( groups6591440286371151544t_real(G,S2) = groups6591440286371151544t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8009_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_int(G,X5) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,S2) = groups4538972089207619220nt_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8010_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) )
               => ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8011_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_real(G,(A3)) = zero_zero_real ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_real(H,(B3)) = zero_zero_real ) )
             => ( ( groups8691415230153176458o_real(G,C2) = groups8691415230153176458o_real(H,C2) )
               => ( groups8691415230153176458o_real(G,A4) = groups8691415230153176458o_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8012_sum_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C2) = groups8778361861064173332t_real(H,C2) )
               => ( groups8778361861064173332t_real(G,A4) = groups8778361861064173332t_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8013_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) )
               => ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8014_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) )
               => ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8015_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_rat,H: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_rat(G,(A3)) = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_rat(H,(B3)) = zero_zero_rat ) )
             => ( ( groups7872700643590313910_o_rat(G,C2) = groups7872700643590313910_o_rat(H,C2) )
               => ( groups7872700643590313910_o_rat(G,A4) = groups7872700643590313910_o_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8016_sum_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C2) = groups3906332499630173760nt_rat(H,C2) )
               => ( groups3906332499630173760nt_rat(G,A4) = groups3906332499630173760nt_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8017_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) )
               => ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8018_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) )
               => ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8019_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_nat(H,(B3)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(G,C2) = groups8507830703676809646_o_nat(H,C2) )
               => ( groups8507830703676809646_o_nat(G,A4) = groups8507830703676809646_o_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8020_sum_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8021_sum_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_real(G,(A3)) = zero_zero_real ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_real(H,(B3)) = zero_zero_real ) )
             => ( ( groups8691415230153176458o_real(G,A4) = groups8691415230153176458o_real(H,B4) )
              <=> ( groups8691415230153176458o_real(G,C2) = groups8691415230153176458o_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8022_sum_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A4) = groups8778361861064173332t_real(H,B4) )
              <=> ( groups8778361861064173332t_real(G,C2) = groups8778361861064173332t_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8023_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8024_sum_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8025_sum_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_rat,H: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_rat(G,(A3)) = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_rat(H,(B3)) = zero_zero_rat ) )
             => ( ( groups7872700643590313910_o_rat(G,A4) = groups7872700643590313910_o_rat(H,B4) )
              <=> ( groups7872700643590313910_o_rat(G,C2) = groups7872700643590313910_o_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8026_sum_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A4) = groups3906332499630173760nt_rat(H,B4) )
              <=> ( groups3906332499630173760nt_rat(G,C2) = groups3906332499630173760nt_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8027_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) )
              <=> ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8028_sum_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) )
              <=> ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8029_sum_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_nat(H,(B3)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(G,A4) = groups8507830703676809646_o_nat(H,B4) )
              <=> ( groups8507830703676809646_o_nat(G,C2) = groups8507830703676809646_o_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8030_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8031_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8032_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8033_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8034_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups2906978787729119204at_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8035_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups3539618377306564664at_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups3539618377306564664at_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8036_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8037_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups7754918857620584856omplex(G,A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,minus_811609699411566653omplex(A4,B4))),groups7754918857620584856omplex(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8038_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups6591440286371151544t_real(G,A4) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups6591440286371151544t_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8039_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4538972089207619220nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A4,B4))),groups4538972089207619220nt_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8040_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,B4)) = aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8041_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8042_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8043_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8044_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8045_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A4,B4)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8046_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_real_real(minus_minus_real(groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8047_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8048_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
       => ( groups4567486121110086003t_real(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_real_real(minus_minus_real(groups4567486121110086003t_real(F,A4)),groups4567486121110086003t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8049_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
       => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups342789780944988191at_rat(F,A4)),groups342789780944988191at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_8050_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,sup_sup_set_complex(A4,B4))),groups5808333547571424918x_real(G,inf_inf_set_complex(A4,B4))) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,A4)),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8051_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,sup_sup_set_complex(A4,B4))),groups5058264527183730370ex_rat(G,inf_inf_set_complex(A4,B4))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,A4)),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8052_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,sup_sup_set_complex(A4,B4))),groups5693394587270226106ex_nat(G,inf_inf_set_complex(A4,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,A4)),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8053_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,sup_sup_set_complex(A4,B4))),groups5690904116761175830ex_int(G,inf_inf_set_complex(A4,B4))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,A4)),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8054_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,A4)),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8055_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,A4)),groups3539618377306564664at_int(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8056_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) ) ) ) ).

% sum.union_inter
tff(fact_8057_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,sup_sup_set_complex(A4,B4))),groups7754918857620584856omplex(G,inf_inf_set_complex(A4,B4))) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A4)),groups7754918857620584856omplex(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8058_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,A4)),groups6591440286371151544t_real(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8059_sum_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,sup_sup_set_int(A4,B4))),groups4538972089207619220nt_int(G,inf_inf_set_int(A4,B4))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A4)),groups4538972089207619220nt_int(G,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8060_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: fun_complex_real,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,A4) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,inf_inf_set_complex(A4,B4))),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8061_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: fun_complex_rat,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,inf_inf_set_complex(A4,B4))),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8062_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: fun_complex_nat,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,inf_inf_set_complex(A4,B4))),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8063_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: fun_complex_int,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,inf_inf_set_complex(A4,B4))),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8064_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: fun_nat_rat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8065_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: fun_nat_int,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups3539618377306564664at_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8066_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: fun_nat_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8067_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: fun_complex_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups7754918857620584856omplex(G,A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,inf_inf_set_complex(A4,B4))),groups7754918857620584856omplex(G,minus_811609699411566653omplex(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8068_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: fun_nat_real,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6591440286371151544t_real(G,A4) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8069_sum_OInt__Diff,axiom,
    ! [A4: set_int,G: fun_int_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4538972089207619220nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,inf_inf_set_int(A4,B4))),groups4538972089207619220nt_int(G,minus_minus_set_int(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8070_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(plus_plus_real(R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_8071_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_8072_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_8073_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_8074_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K2: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8075_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K2: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8076_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K2: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8077_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8078_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8079_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8080_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8081_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8082_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8083_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8084_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8085_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8086_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8087_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8088_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8089_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8090_powser__inside,axiom,
    ! [F: fun_nat_real,X3: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),X3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X3))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z)) ) ) ).

% powser_inside
tff(fact_8091_powser__inside,axiom,
    ! [F: fun_nat_complex,X3: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),X3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X3))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z)) ) ) ).

% powser_inside
tff(fact_8092_summable__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => summable_real(power_power_real(C)) ) ).

% summable_geometric
tff(fact_8093_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_8094_complete__algebra__summable__geometric,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),one_one_real)
     => summable_real(power_power_real(X3)) ) ).

% complete_algebra_summable_geometric
tff(fact_8095_complete__algebra__summable__geometric,axiom,
    ! [X3: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),one_one_real)
     => summable_complex(power_power_complex(X3)) ) ).

% complete_algebra_summable_geometric
tff(fact_8096_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_il(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8097_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: fun_complex_o,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_jg(P),H),G),A4) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(H,inf_inf_set_complex(A4,collect_complex(P)))),groups5808333547571424918x_real(G,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8098_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: fun_complex_o,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_jh(P),H),G),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(H,inf_inf_set_complex(A4,collect_complex(P)))),groups5058264527183730370ex_rat(G,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8099_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: fun_complex_o,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ji(P),H),G),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(H,inf_inf_set_complex(A4,collect_complex(P)))),groups5693394587270226106ex_nat(G,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8100_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: fun_complex_o,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_jj(P),H),G),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(H,inf_inf_set_complex(A4,collect_complex(P)))),groups5690904116761175830ex_int(G,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8101_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: fun_nat_o,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_jk(P),H),G),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(H,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P)))),groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8102_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: fun_nat_o,H: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jl(P),H),G),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(H,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P)))),groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8103_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: fun_nat_o,H: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jm(P),H),G)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(H),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P)))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8104_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: fun_complex_o,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aa_fun4268748527592579270omplex(aTP_Lamm_jn(P),H),G),A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(H,inf_inf_set_complex(A4,collect_complex(P)))),groups7754918857620584856omplex(G,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8105_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: fun_nat_o,H: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_jo(P),H),G),A4) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(H,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P)))),groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(P))))) ) ) ).

% sum.If_cases
tff(fact_8106_sum_OIf__cases,axiom,
    ! [A4: set_int,P: fun_int_o,H: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_jp(P),H),G),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(H,inf_inf_set_int(A4,collect_int(P)))),groups4538972089207619220nt_int(G,inf_inf_set_int(A4,uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8107_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8108_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8109_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8110_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),groups6591440286371151544t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8111_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_js(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_8112_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_jt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_8113_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(aTP_Lamm_ju(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_8114_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8115_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(B3))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8691415230153176458o_real(F,A4)),groups8691415230153176458o_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8116_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8117_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8118_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8119_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(B3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,A4)),groups7872700643590313910_o_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8120_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8121_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8122_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8123_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(B3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,A4)),groups8507830703676809646_o_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_8124_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),inf_inf_set_complex(A4,B4))
             => ( aa_complex_real(G,X5) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,sup_sup_set_complex(A4,B4)) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,A4)),groups5808333547571424918x_real(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8125_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),inf_inf_set_complex(A4,B4))
             => ( aa_complex_rat(G,X5) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,sup_sup_set_complex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,A4)),groups5058264527183730370ex_rat(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8126_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),inf_inf_set_complex(A4,B4))
             => ( aa_complex_nat(G,X5) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,sup_sup_set_complex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,A4)),groups5693394587270226106ex_nat(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8127_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),inf_inf_set_complex(A4,B4))
             => ( aa_complex_int(G,X5) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,sup_sup_set_complex(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,A4)),groups5690904116761175830ex_int(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8128_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_rat(G,X5) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,A4)),groups2906978787729119204at_rat(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8129_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_int(G,X5) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,A4)),groups3539618377306564664at_int(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8130_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_nat(G,X5) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8131_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),inf_inf_set_complex(A4,B4))
             => ( aa_complex_complex(G,X5) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G,sup_sup_set_complex(A4,B4)) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A4)),groups7754918857620584856omplex(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8132_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_real(G,X5) = zero_zero_real ) )
         => ( groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,A4)),groups6591440286371151544t_real(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8133_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),inf_inf_set_int(A4,B4))
             => ( aa_int_int(G,X5) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,sup_sup_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A4)),groups4538972089207619220nt_int(G,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8134_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_real,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,insert_complex(X3,A4)) = aa_real_real(plus_plus_real(aa_complex_real(G,X3)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8135_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_rat,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,insert_complex(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X3)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8136_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_nat,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5693394587270226106ex_nat(G,insert_complex(X3,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X3)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8137_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_int,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5690904116761175830ex_int(G,insert_complex(X3,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X3)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8138_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_real,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8691415230153176458o_real(G,insert_o((X3),A4)) = aa_real_real(plus_plus_real(aa_o_real(G,(X3))),groups8691415230153176458o_real(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8139_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_rat,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(G,insert_o((X3),A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X3))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8140_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_nat,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8507830703676809646_o_nat(G,insert_o((X3),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(X3))),groups8507830703676809646_o_nat(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8141_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_int,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8505340233167759370_o_int(G,insert_o((X3),A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(G,(X3))),groups8505340233167759370_o_int(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8142_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_real,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,insert_int(X3,A4)) = aa_real_real(plus_plus_real(aa_int_real(G,X3)),groups8778361861064173332t_real(G,minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8143_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_rat,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,insert_int(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X3)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8144_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => ( groups8097168146408367636l_real(G,A4) = aa_real_real(plus_plus_real(aa_real_real(G,X3)),groups8097168146408367636l_real(G,minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8145_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(plus_plus_real(aa_complex_real(G,X3)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8146_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => ( groups1300246762558778688al_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X3)),groups1300246762558778688al_rat(G,minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8147_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X3)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8148_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => ( groups1935376822645274424al_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X3)),groups1935376822645274424al_nat(G,minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8149_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X3)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8150_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => ( groups1932886352136224148al_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X3)),groups1932886352136224148al_int(G,minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8151_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X3)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8152_sum_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X3)),A4)
       => ( groups8691415230153176458o_real(G,A4) = aa_real_real(plus_plus_real(aa_o_real(G,(X3))),groups8691415230153176458o_real(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8153_sum_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X3)),A4)
       => ( groups7872700643590313910_o_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X3))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8154_sum__diff1,axiom,
    ! [A4: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups8097168146408367636l_real(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_real_real(minus_minus_real(groups8097168146408367636l_real(F,A4)),aa_real_real(F,A)),groups8097168146408367636l_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_8155_sum__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A4)),aa_complex_real(F,A)),groups5808333547571424918x_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_8156_sum__diff1,axiom,
    ! [A4: set_o,F: fun_o_real,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8691415230153176458o_real(F,minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),aa_real_real(minus_minus_real(groups8691415230153176458o_real(F,A4)),aa_o_real(F,(A))),groups8691415230153176458o_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_8157_sum__diff1,axiom,
    ! [A4: set_int,F: fun_int_real,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),aa_real_real(minus_minus_real(groups8778361861064173332t_real(F,A4)),aa_int_real(F,A)),groups8778361861064173332t_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_8158_sum__diff1,axiom,
    ! [A4: set_real,F: fun_real_rat,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups1300246762558778688al_rat(F,A4)),aa_real_rat(F,A)),groups1300246762558778688al_rat(F,A4)) ) ) ).

% sum_diff1
tff(fact_8159_sum__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5058264527183730370ex_rat(F,A4)),aa_complex_rat(F,A)),groups5058264527183730370ex_rat(F,A4)) ) ) ).

% sum_diff1
tff(fact_8160_sum__diff1,axiom,
    ! [A4: set_o,F: fun_o_rat,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(F,minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups7872700643590313910_o_rat(F,A4)),aa_o_rat(F,(A))),groups7872700643590313910_o_rat(F,A4)) ) ) ).

% sum_diff1
tff(fact_8161_sum__diff1,axiom,
    ! [A4: set_int,F: fun_int_rat,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(F,A4)),aa_int_rat(F,A)),groups3906332499630173760nt_rat(F,A4)) ) ) ).

% sum_diff1
tff(fact_8162_sum__diff1,axiom,
    ! [A4: set_real,F: fun_real_int,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups1932886352136224148al_int(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(F,A4)),aa_real_int(F,A)),groups1932886352136224148al_int(F,A4)) ) ) ).

% sum_diff1
tff(fact_8163_sum__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_int,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),aa_complex_int(F,A)),groups5690904116761175830ex_int(F,A4)) ) ) ).

% sum_diff1
tff(fact_8164_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5808333547571424918x_real(F,sup_sup_set_complex(A4,B4)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4))),groups5808333547571424918x_real(F,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8165_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5058264527183730370ex_rat(F,sup_sup_set_complex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4))),groups5058264527183730370ex_rat(F,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8166_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4))),groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8167_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5690904116761175830ex_int(F,sup_sup_set_complex(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4))),groups5690904116761175830ex_int(F,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8168_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(F,B4))),groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8169_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups7754918857620584856omplex(F,sup_sup_set_complex(A4,B4)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(F,B4))),groups7754918857620584856omplex(F,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8170_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(F,B4))),groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8171_sum__Un,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4538972089207619220nt_int(F,sup_sup_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(F,B4))),groups4538972089207619220nt_int(F,inf_inf_set_int(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8172_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups4567486121110086003t_real(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(groups4567486121110086003t_real(F,A4)),groups4567486121110086003t_real(F,B4))),groups4567486121110086003t_real(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8173_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(F,A4)),groups342789780944988191at_rat(F,B4))),groups342789780944988191at_rat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8174_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups5808333547571424918x_real(G,sup_sup_set_complex(A4,B4)) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,A4)),groups5808333547571424918x_real(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8175_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups5058264527183730370ex_rat(G,sup_sup_set_complex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,A4)),groups5058264527183730370ex_rat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8176_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups5693394587270226106ex_nat(G,sup_sup_set_complex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,A4)),groups5693394587270226106ex_nat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8177_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups5690904116761175830ex_int(G,sup_sup_set_complex(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,A4)),groups5690904116761175830ex_int(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8178_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups8691415230153176458o_real(G,sup_sup_set_o(A4,B4)) = aa_real_real(plus_plus_real(groups8691415230153176458o_real(G,A4)),groups8691415230153176458o_real(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8179_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups7872700643590313910_o_rat(G,sup_sup_set_o(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups7872700643590313910_o_rat(G,A4)),groups7872700643590313910_o_rat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8180_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups8507830703676809646_o_nat(G,sup_sup_set_o(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8507830703676809646_o_nat(G,A4)),groups8507830703676809646_o_nat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8181_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups8505340233167759370_o_int(G,sup_sup_set_o(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups8505340233167759370_o_int(G,A4)),groups8505340233167759370_o_int(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8182_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,A4)),groups2906978787729119204at_rat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8183_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,A4)),groups3539618377306564664at_int(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8184_ceiling__correct,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X3))),one_one_real)),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X3))) ) ).

% ceiling_correct
tff(fact_8185_ceiling__correct,axiom,
    ! [X3: 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,X3))),one_one_rat)),X3)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X3))) ) ).

% ceiling_correct
tff(fact_8186_ceiling__unique,axiom,
    ! [Z: int,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,X3) = Z ) ) ) ).

% ceiling_unique
tff(fact_8187_ceiling__unique,axiom,
    ! [Z: int,X3: 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)),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X3) = Z ) ) ) ).

% ceiling_unique
tff(fact_8188_ceiling__eq__iff,axiom,
    ! [X3: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X3) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)),X3)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_8189_ceiling__eq__iff,axiom,
    ! [X3: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X3) = 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)),X3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_8190_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(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_8191_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_8192_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A4,B4))
     => ( groups5808333547571424918x_real(F,sup_sup_set_complex(A4,B4)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(F,minus_811609699411566653omplex(B4,A4)))),groups5808333547571424918x_real(F,inf_inf_set_complex(A4,B4))) ) ) ).

% sum_Un2
tff(fact_8193_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A4,B4))
     => ( groups5058264527183730370ex_rat(F,sup_sup_set_complex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(B4,A4)))),groups5058264527183730370ex_rat(F,inf_inf_set_complex(A4,B4))) ) ) ).

% sum_Un2
tff(fact_8194_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A4,B4))
     => ( groups5693394587270226106ex_nat(F,sup_sup_set_complex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(B4,A4)))),groups5693394587270226106ex_nat(F,inf_inf_set_complex(A4,B4))) ) ) ).

% sum_Un2
tff(fact_8195_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A4,B4))
     => ( groups5690904116761175830ex_int(F,sup_sup_set_complex(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(F,minus_811609699411566653omplex(B4,A4)))),groups5690904116761175830ex_int(F,inf_inf_set_complex(A4,B4))) ) ) ).

% sum_Un2
tff(fact_8196_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_8197_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_8198_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_8199_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A4,B4))
     => ( groups7754918857620584856omplex(F,sup_sup_set_complex(A4,B4)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(F,minus_811609699411566653omplex(A4,B4))),groups7754918857620584856omplex(F,minus_811609699411566653omplex(B4,A4)))),groups7754918857620584856omplex(F,inf_inf_set_complex(A4,B4))) ) ) ).

% sum_Un2
tff(fact_8200_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups6591440286371151544t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_8201_sum__Un2,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,sup_sup_set_int(A4,B4))
     => ( groups4538972089207619220nt_int(F,sup_sup_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(F,minus_minus_set_int(A4,B4))),groups4538972089207619220nt_int(F,minus_minus_set_int(B4,A4)))),groups4538972089207619220nt_int(F,inf_inf_set_int(A4,B4))) ) ) ).

% sum_Un2
tff(fact_8202_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5808333547571424918x_real(G,sup_sup_set_complex(A4,B4)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(G,minus_811609699411566653omplex(B4,A4)))),groups5808333547571424918x_real(G,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8203_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5058264527183730370ex_rat(G,sup_sup_set_complex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(B4,A4)))),groups5058264527183730370ex_rat(G,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8204_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5693394587270226106ex_nat(G,sup_sup_set_complex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(B4,A4)))),groups5693394587270226106ex_nat(G,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8205_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5690904116761175830ex_int(G,sup_sup_set_complex(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(B4,A4)))),groups5690904116761175830ex_int(G,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8206_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8207_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups3539618377306564664at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8208_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8209_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups7754918857620584856omplex(G,sup_sup_set_complex(A4,B4)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,minus_811609699411566653omplex(A4,B4))),groups7754918857620584856omplex(G,minus_811609699411566653omplex(B4,A4)))),groups7754918857620584856omplex(G,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8210_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups6591440286371151544t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8211_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4538972089207619220nt_int(G,sup_sup_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A4,B4))),groups4538972089207619220nt_int(G,minus_minus_set_int(B4,A4)))),groups4538972089207619220nt_int(G,inf_inf_set_int(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_8212_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_8213_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).

% mult_ceiling_le
tff(fact_8214_ceiling__less__iff,axiom,
    ! [X3: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X3)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_8215_ceiling__less__iff,axiom,
    ! [X3: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X3)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),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_8216_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8217_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8218_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8219_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8220_sum__div__partition,axiom,
    ! [A4: set_complex,F: fun_co8153928105989009288nteger,B: code_integer] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( divide6298287555418463151nteger(groups6621422865394947399nteger(F,A4),B) = plus_p5714425477246183910nteger(groups6621422865394947399nteger(aa_Cod1660996421839551523nteger(aTP_Lamm_jv(F),B),inf_inf_set_complex(A4,collect_complex(aa_Cod7908732550325689320plex_o(aTP_Lamm_jw(F),B)))),divide6298287555418463151nteger(groups6621422865394947399nteger(F,inf_inf_set_complex(A4,collect_complex(aa_Cod7908732550325689320plex_o(aTP_Lamm_jx(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8221_sum__div__partition,axiom,
    ! [A4: set_nat,F: fun_nat_Code_integer,B: code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( divide6298287555418463151nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(F),A4),B) = plus_p5714425477246183910nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_jy(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_jz(F),B)))),divide6298287555418463151nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_ka(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8222_sum__div__partition,axiom,
    ! [A4: set_complex,F: fun_complex_nat,B: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups5693394587270226106ex_nat(F,A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_kb(F),B),inf_inf_set_complex(A4,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_kc(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups5693394587270226106ex_nat(F,inf_inf_set_complex(A4,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_kd(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8223_sum__div__partition,axiom,
    ! [A4: set_complex,F: fun_complex_int,B: int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups5690904116761175830ex_int(F,A4)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ke(F),B),inf_inf_set_complex(A4,collect_complex(aa_int_fun_complex_o(aTP_Lamm_kf(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups5690904116761175830ex_int(F,inf_inf_set_complex(A4,collect_complex(aa_int_fun_complex_o(aTP_Lamm_kg(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8224_sum__div__partition,axiom,
    ! [A4: set_nat,F: fun_nat_int,B: int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups3539618377306564664at_int(F,A4)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_kh(F),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_int_fun_nat_o(aTP_Lamm_ki(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_int_fun_nat_o(aTP_Lamm_kj(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8225_sum__div__partition,axiom,
    ! [A4: set_nat,F: fun_nat_nat,B: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kk(F),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_kl(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_km(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8226_sum__div__partition,axiom,
    ! [A4: set_int,F: fun_int_int,B: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups4538972089207619220nt_int(F,A4)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_kn(F),B),inf_inf_set_int(A4,collect_int(aa_int_fun_int_o(aTP_Lamm_ko(F),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups4538972089207619220nt_int(F,inf_inf_set_int(A4,collect_int(aa_int_fun_int_o(aTP_Lamm_kp(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8227_sum__div__partition,axiom,
    ! [A4: set_set_nat,F: fun_se7586849273513198688nteger,B: code_integer] :
      ( finite1152437895449049373et_nat(A4)
     => ( divide6298287555418463151nteger(groups9190459664516455967nteger(F,A4),B) = plus_p5714425477246183910nteger(groups9190459664516455967nteger(aa_Cod3249490220333259515nteger(aTP_Lamm_kq(F),B),inf_inf_set_set_nat(A4,collect_set_nat(aa_Cod793699401815963408_nat_o(aTP_Lamm_kr(F),B)))),divide6298287555418463151nteger(groups9190459664516455967nteger(F,inf_inf_set_set_nat(A4,collect_set_nat(aa_Cod793699401815963408_nat_o(aTP_Lamm_ks(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8228_sum__div__partition,axiom,
    ! [A4: set_list_nat,F: fun_li3231056112175589690nteger,B: code_integer] :
      ( finite8100373058378681591st_nat(A4)
     => ( divide6298287555418463151nteger(groups3257113351181983737nteger(F,A4),B) = plus_p5714425477246183910nteger(groups3257113351181983737nteger(aa_Cod1658353545975407189nteger(aTP_Lamm_kt(F),B),inf_inf_set_list_nat(A4,collect_list_nat(aa_Cod7157825952861150518_nat_o(aTP_Lamm_ku(F),B)))),divide6298287555418463151nteger(groups3257113351181983737nteger(F,inf_inf_set_list_nat(A4,collect_list_nat(aa_Cod7157825952861150518_nat_o(aTP_Lamm_kv(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8229_sum__div__partition,axiom,
    ! [A4: set_set_nat,F: fun_set_nat_nat,B: nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups8294997508430121362at_nat(F,A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_kw(F),B),inf_inf_set_set_nat(A4,collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_kx(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups8294997508430121362at_nat(F,inf_inf_set_set_nat(A4,collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_ky(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8230_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_kz(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(plus_plus_real(aa_real_real(B,A)),groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8231_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_la(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_real_real(plus_plus_real(aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8232_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lb(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(B,A)),groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8233_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_lc(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8234_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat,C: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_ld(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B,A)),groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8235_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_le(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8236_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_int,C: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_lf(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(B,A)),groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8237_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lg(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8238_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: fun_o_real,C: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( groups8691415230153176458o_real(aa_fun688608109223240316o_real(aa_fun1133276423545935902o_real(aTP_Lamm_lh((A)),B),C),S2) = $ite(aa_set_o_o(member_o((A)),S2),aa_real_real(plus_plus_real(aa_o_real(B,(A))),groups8691415230153176458o_real(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))),groups8691415230153176458o_real(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8239_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: fun_o_rat,C: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_li((A)),B),C),S2) = $ite(aa_set_o_o(member_o((A)),S2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(B,(A))),groups7872700643590313910_o_rat(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))),groups7872700643590313910_o_rat(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8240_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_8241_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X5: real] :
                  ( aa_set_real_o(member_real(X5),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8242_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(B)))
           => ( ! [X5: $o] :
                  ( aa_set_o_o(member_o((X5)),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X5))) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8691415230153176458o_real(F,A4)),groups8691415230153176458o_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8243_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( aa_set_int_o(member_int(B),minus_minus_set_int(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,B))
           => ( ! [X5: int] :
                  ( aa_set_int_o(member_int(X5),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8244_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X5: complex] :
                  ( aa_set_complex_o(member_complex(X5),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8245_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X5: real] :
                  ( aa_set_real_o(member_real(X5),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8246_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(B)))
           => ( ! [X5: $o] :
                  ( aa_set_o_o(member_o((X5)),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X5))) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A4)),groups7872700643590313910_o_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8247_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( aa_set_int_o(member_int(B),minus_minus_set_int(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,B))
           => ( ! [X5: int] :
                  ( aa_set_int_o(member_int(X5),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8248_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X5: complex] :
                  ( aa_set_complex_o(member_complex(X5),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8249_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X5: real] :
                  ( aa_set_real_o(member_real(X5),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X5)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8250_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(B)))
           => ( ! [X5: $o] :
                  ( aa_set_o_o(member_o((X5)),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X5))) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A4)),groups8507830703676809646_o_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8251_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real(I),A4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),minus_minus_set_real(A4,insert_real(I,bot_bot_set_real)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5)) )
       => ( aa_set_real_o(finite_finite_real,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8252_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(I),A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(A4,insert_complex(I,bot_bot_set_complex)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8253_member__le__sum,axiom,
    ! [I: $o,A4: set_o,F: fun_o_real] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),minus_minus_set_o(A4,insert_o((I),bot_bot_set_o)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X5))) )
       => ( aa_set_o_o(finite_finite_o,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I))),groups8691415230153176458o_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8254_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(member_int(I),A4)
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),minus_minus_set_int(A4,insert_int(I,bot_bot_set_int)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5)) )
       => ( aa_set_int_o(finite_finite_int,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8255_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(member_real(I),A4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),minus_minus_set_real(A4,insert_real(I,bot_bot_set_real)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5)) )
       => ( aa_set_real_o(finite_finite_real,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8256_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(I),A4)
     => ( ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(A4,insert_complex(I,bot_bot_set_complex)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8257_member__le__sum,axiom,
    ! [I: $o,A4: set_o,F: fun_o_rat] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),minus_minus_set_o(A4,insert_o((I),bot_bot_set_o)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X5))) )
       => ( aa_set_o_o(finite_finite_o,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I))),groups7872700643590313910_o_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8258_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(I),A4)
     => ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),minus_minus_set_int(A4,insert_int(I,bot_bot_set_int)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5)) )
       => ( aa_set_int_o(finite_finite_int,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8259_member__le__sum,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(I),A4)
     => ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(I,bot_bot_set_nat)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5)) )
       => ( aa_set_nat_o(finite_finite_nat,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8260_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: fun_real_nat] :
      ( aa_set_real_o(member_real(I),A4)
     => ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),minus_minus_set_real(A4,insert_real(I,bot_bot_set_real)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X5)) )
       => ( aa_set_real_o(finite_finite_real,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),groups1935376822645274424al_nat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_8261_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_8262_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),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,P2),Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_8263_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8264_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z)) = aa_real_real(plus_plus_real(aa_nat_real(F,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_iw(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8265_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8266_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_iw(F),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8267_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_real(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8268_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_complex(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_lj(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8269_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X3,I3)) )
     => ( ( groups6621422865394947399nteger(X3,I5) = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_lk(X3),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8270_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X3,I3)) )
     => ( ( groups7713935264441627589nteger(X3,I5) = one_one_Code_integer )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_ll(X3),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8271_convex__sum__bound__le,axiom,
    ! [I5: set_o,X3: fun_o_Code_integer,A: fun_o_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(X3,(I3))) )
     => ( ( groups4406642042086082107nteger(X3,I5) = one_one_Code_integer )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_o_Code_integer(A,(I3)),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_lm(X3),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8272_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X3,I3)) )
     => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(X3),I5) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ln(X3),A)),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8273_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X3,I3)) )
     => ( ( groups7873554091576472773nteger(X3,I5) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_lo(X3),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8274_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(X3,I3)) )
     => ( ( groups5808333547571424918x_real(X3,I5) = one_one_real )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(A,I3)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_lp(X3),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8275_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(X3,I3)) )
     => ( ( groups8097168146408367636l_real(X3,I5) = one_one_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(A,I3)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_lq(X3),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8276_convex__sum__bound__le,axiom,
    ! [I5: set_o,X3: fun_o_real,A: fun_o_real,B: real,Delta: real] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(X3,(I3))) )
     => ( ( groups8691415230153176458o_real(X3,I5) = one_one_real )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_o_real(A,(I3))),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8691415230153176458o_real(aa_fun688608109223240316o_real(aTP_Lamm_lr(X3),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8277_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(X3,I3)) )
     => ( ( groups8778361861064173332t_real(X3,I5) = one_one_real )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(A,I3)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ls(X3),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8278_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X3,I3)) )
     => ( ( groups5058264527183730370ex_rat(X3,I5) = one_one_rat )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => 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_complex_rat(A,I3)),B))),Delta) )
         => 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,groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_lt(X3),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8279_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8280_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M6: 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_real,R2),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(power_power_real(R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_lu(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_8281_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_lv(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,Mb)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_8282_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_lw(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_8283_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(aTP_Lamm_lx(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_8284_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_ly(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_8285_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_lz(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_8286_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6591440286371151544t_real(aTP_Lamm_ma(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_8287_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8288_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8289_ceiling__eq,axiom,
    ! [Nb: int,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,X3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_8290_ceiling__eq,axiom,
    ! [Nb: int,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_8291_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Nb))) ).

% mask_eq_sum_exp
tff(fact_8292_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Nb))) ).

% mask_eq_sum_exp
tff(fact_8293_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,X3: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X3)),groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X3),Mb)),aa_nat_complex(power_power_complex(X3),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8294_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,X3: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3)),groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X3),Mb)),aa_nat_rat(power_power_rat(X3),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8295_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,X3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X3)),groups3539618377306564664at_int(power_power_int(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X3),Mb)),aa_nat_int(power_power_int(X3),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8296_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X3)),groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),Mb)),aa_nat_real(power_power_real(X3),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8297_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_mc(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_8298_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_md(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_8299_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_me(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_8300_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_mf(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_8301_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_8302_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8303_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_8304_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8305_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_8306_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_8307_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_8308_double__arith__series,axiom,
    ! [A: complex,D: complex,Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mh(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),D))) ).

% double_arith_series
tff(fact_8309_double__arith__series,axiom,
    ! [A: rat,D: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_mi(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),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)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).

% double_arith_series
tff(fact_8310_double__arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_mj(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_8311_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_8312_double__arith__series,axiom,
    ! [A: real,D: real,Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ml(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_8313_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mm(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8314_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8315_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_mj(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8316_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8317_gauss__sum,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8318_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8319_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8320_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8321_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8322_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8323_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8324_sum__gp__offset,axiom,
    ! [X3: complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(X3 = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X3),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X3),aa_nat_nat(suc,Nb))))),aa_complex_complex(minus_minus_complex(one_one_complex),X3))) ).

% sum_gp_offset
tff(fact_8325_sum__gp__offset,axiom,
    ! [X3: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(X3 = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),Mb)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X3),aa_nat_nat(suc,Nb))))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3))) ).

% sum_gp_offset
tff(fact_8326_sum__gp__offset,axiom,
    ! [X3: real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(X3 = one_one_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(X3),Mb)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_nat_nat(suc,Nb))))),aa_real_real(minus_minus_real(one_one_real),X3))) ).

% sum_gp_offset
tff(fact_8327_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8328_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8329_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mo(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8330_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: nat] :
      ( ( H != 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,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),H)),Nb)),aa_nat_rat(power_power_rat(Z),Nb))),H)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mq(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8331_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(power_power_real(Z),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(times_times_real(H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ms(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8332_ceiling__log__eq__powr__iff,axiom,
    ! [X3: real,B: real,K2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B),X3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K2)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8333_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aTP_Lamm_mt(Z)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8334_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => aa_complex_o(sums_complex(aTP_Lamm_mu(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8335_monoseq__def,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,M4)) ) ) ) ).

% monoseq_def
tff(fact_8336_monoseq__def,axiom,
    ! [X6: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M4)),aa_nat_set_nat(X6,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N3)),aa_nat_set_nat(X6,M4)) ) ) ) ).

% monoseq_def
tff(fact_8337_monoseq__def,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M4)),aa_nat_rat(X6,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,M4)) ) ) ) ).

% monoseq_def
tff(fact_8338_monoseq__def,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M4)),aa_nat_num(X6,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,M4)) ) ) ) ).

% monoseq_def
tff(fact_8339_monoseq__def,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M4)),aa_nat_nat(X6,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,M4)) ) ) ) ).

% monoseq_def
tff(fact_8340_monoseq__def,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M4)),aa_nat_int(X6,N3)) )
        | ! [M4: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,M4)) ) ) ) ).

% monoseq_def
tff(fact_8341_monoI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,M)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI2
tff(fact_8342_monoI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,M)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI2
tff(fact_8343_monoI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,M)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI2
tff(fact_8344_monoI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,M)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI2
tff(fact_8345_monoI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,M)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI2
tff(fact_8346_monoI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,M)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI2
tff(fact_8347_lessThan__subset__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(X3)),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ).

% lessThan_subset_iff
tff(fact_8348_lessThan__subset__iff,axiom,
    ! [X3: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(X3)),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y) ) ).

% lessThan_subset_iff
tff(fact_8349_lessThan__subset__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(X3)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ).

% lessThan_subset_iff
tff(fact_8350_lessThan__subset__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X3)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ).

% lessThan_subset_iff
tff(fact_8351_lessThan__subset__iff,axiom,
    ! [X3: real,Y: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or5984915006950818249n_real(X3)),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y) ) ).

% lessThan_subset_iff
tff(fact_8352_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8353_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8354_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8355_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8356_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_8357_single__Diff__lessThan,axiom,
    ! [K2: $o] : minus_minus_set_o(insert_o((K2),bot_bot_set_o),set_ord_lessThan_o((K2))) = insert_o((K2),bot_bot_set_o) ).

% single_Diff_lessThan
tff(fact_8358_single__Diff__lessThan,axiom,
    ! [K2: int] : minus_minus_set_int(insert_int(K2,bot_bot_set_int),set_ord_lessThan_int(K2)) = insert_int(K2,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_8359_single__Diff__lessThan,axiom,
    ! [K2: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(K2,bot_bot_set_nat)),aa_nat_set_nat(set_ord_lessThan_nat,K2)) = insert_nat(K2,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_8360_single__Diff__lessThan,axiom,
    ! [K2: real] : minus_minus_set_real(insert_real(K2,bot_bot_set_real),set_or5984915006950818249n_real(K2)) = insert_real(K2,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_8361_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(powr_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_8362_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X3: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_hy(A)),X3)
    <=> ( aa_nat_complex(A,zero_zero_nat) = X3 ) ) ).

% powser_sums_zero_iff
tff(fact_8363_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X3: real] :
      ( aa_real_o(sums_real(aTP_Lamm_hz(A)),X3)
    <=> ( aa_nat_real(A,zero_zero_nat) = X3 ) ) ).

% powser_sums_zero_iff
tff(fact_8364_powr__numeral,axiom,
    ! [X3: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_real(powr_real(X3),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_8365_square__powr__half,axiom,
    ! [X3: real] : aa_real_real(powr_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X3) ).

% square_powr_half
tff(fact_8366_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Sb: real,Ta: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( aa_real_o(sums_real(F),Sb)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Sb),Ta) ) ) ) ).

% sums_le
tff(fact_8367_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Sb: nat,Ta: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( aa_nat_o(sums_nat(F),Sb)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Sb),Ta) ) ) ) ).

% sums_le
tff(fact_8368_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,Sb: int,Ta: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( aa_int_o(sums_int(F),Sb)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Sb),Ta) ) ) ) ).

% sums_le
tff(fact_8369_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Nb: real] :
      ( ! [X5: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X5)),aa_real_nat(P,X5))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_mv(Q),P),set_or5984915006950818249n_real(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8370_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
      ( ! [X5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X5)),aa_nat_nat(P,X5))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mw(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_8371_lessThan__non__empty,axiom,
    ! [X3: int] : set_ord_lessThan_int(X3) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_8372_lessThan__non__empty,axiom,
    ! [X3: real] : set_or5984915006950818249n_real(X3) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_8373_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(F),G)),aa_real_real(plus_plus_real(A),B)) ) ) ).

% sums_add
tff(fact_8374_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_8375_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ik(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_8376_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,Sb: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),Nb)),Sb)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(Sb),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_8377_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,Sb: real,Nb: nat] :
      ( aa_real_o(sums_real(F),Sb)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),Nb)),aa_real_real(minus_minus_real(Sb),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_8378_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,Sb: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),Nb)),aa_real_real(minus_minus_real(Sb),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_real_o(sums_real(F),Sb) ) ).

% sums_iff_shift'
tff(fact_8379_Iio__eq__empty__iff,axiom,
    ! [Nb: $o] :
      ( ( set_ord_lessThan_o((Nb)) = bot_bot_set_o )
    <=> ( (Nb)
      <=> bot_bot_o ) ) ).

% Iio_eq_empty_iff
tff(fact_8380_Iio__eq__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_8381_lessThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = insert_nat(K2,aa_nat_set_nat(set_ord_lessThan_nat,K2)) ).

% lessThan_Suc
tff(fact_8382_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8383_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X5)),aa_complex_nat(F,X5)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_mx(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_8384_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X5)),aa_real_nat(F,X5)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_mv(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_8385_sum__subtractf__nat,axiom,
    ! [A4: set_o,G: fun_o_nat,F: fun_o_nat] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(G,(X5))),aa_o_nat(F,(X5))) )
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_my(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A4)),groups8507830703676809646_o_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_8386_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X5)),aa_int_nat(F,X5)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_mz(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_8387_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X5)),aa_nat_nat(F,X5)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mw(G),F)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_8388_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & ( aa_complex_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8389_sum__eq__Suc0__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ( groups977919841031483927at_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & ( aa_Pro4243173770044397973at_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8390_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ( aa_nat_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A4)
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8391_sum__SucD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,Nb) )
     => ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X5)) ) ) ).

% sum_SucD
tff(fact_8392_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,Sb: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_il(F)),Sb)
       => aa_real_o(sums_real(F),Sb) ) ) ).

% sums_Suc_imp
tff(fact_8393_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,Sb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_il(F)),Sb)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(Sb),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8394_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_il(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(plus_plus_real(L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8395_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_is(F)),L)
     => aa_nat_o(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8396_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_jr(F)),L)
     => aa_int_o(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8397_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,Sb: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(Nb),F)),Sb)
      <=> aa_real_o(sums_real(F),Sb) ) ) ).

% sums_zero_iff_shift
tff(fact_8398_ivl__disj__int__one_I4_J,axiom,
    ! [L: $o,U: $o] : inf_inf_set_o(set_ord_lessThan_o((L)),set_or8904488021354931149Most_o((L),(U))) = bot_bot_set_o ).

% ivl_disj_int_one(4)
tff(fact_8399_ivl__disj__int__one_I4_J,axiom,
    ! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = bot_bot_set_nat ).

% ivl_disj_int_one(4)
tff(fact_8400_ivl__disj__int__one_I4_J,axiom,
    ! [L: int,U: int] : inf_inf_set_int(set_ord_lessThan_int(L),set_or1266510415728281911st_int(L,U)) = bot_bot_set_int ).

% ivl_disj_int_one(4)
tff(fact_8401_ivl__disj__int__one_I4_J,axiom,
    ! [L: real,U: real] : inf_inf_set_real(set_or5984915006950818249n_real(L),set_or1222579329274155063t_real(L,U)) = bot_bot_set_real ).

% ivl_disj_int_one(4)
tff(fact_8402_powr__add,axiom,
    ! [X3: real,A: real,B: real] : aa_real_real(powr_real(X3),aa_real_real(plus_plus_real(A),B)) = aa_real_real(times_times_real(aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(X3),B)) ).

% powr_add
tff(fact_8403_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K2))) ).

% lessThan_nat_numeral
tff(fact_8404_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_8405_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nc(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_8406_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_nd,collect_complex(aTP_Lamm_ev(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8407_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_nd,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ew(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8408_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ne(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).

% powser_sums_if
tff(fact_8409_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_nf(Mb),Z)),aa_nat_real(power_power_real(Z),Mb)) ).

% powser_sums_if
tff(fact_8410_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_ng(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).

% powser_sums_if
tff(fact_8411_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_hy(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8412_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_hz(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8413_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8414_sum__diff__nat,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
       => ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),groups977919841031483927at_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8415_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8416_sum__diff1__nat,axiom,
    ! [F: fun_Pr1253225528153422666at_nat,A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups3860910324918113789at_nat(F,A4)),aa_Pro6313114768868464763at_nat(F,A)),groups3860910324918113789at_nat(F,A4)) ).

% sum_diff1_nat
tff(fact_8417_sum__diff1__nat,axiom,
    ! [F: fun_complex_nat,A4: set_complex,A: complex] :
      groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),aa_complex_nat(F,A)),groups5693394587270226106ex_nat(F,A4)) ).

% sum_diff1_nat
tff(fact_8418_sum__diff1__nat,axiom,
    ! [F: fun_real_nat,A4: set_real,A: real] :
      groups1935376822645274424al_nat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),aa_real_nat(F,A)),groups1935376822645274424al_nat(F,A4)) ).

% sum_diff1_nat
tff(fact_8419_sum__diff1__nat,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),aa_Pro4243173770044397973at_nat(F,A)),groups977919841031483927at_nat(F,A4)) ).

% sum_diff1_nat
tff(fact_8420_sum__diff1__nat,axiom,
    ! [F: fun_o_nat,A4: set_o,A: $o] :
      groups8507830703676809646_o_nat(F,minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A4)),aa_o_nat(F,(A))),groups8507830703676809646_o_nat(F,A4)) ).

% sum_diff1_nat
tff(fact_8421_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A4: set_int,A: int] :
      groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),aa_int_nat(F,A)),groups4541462559716669496nt_nat(F,A4)) ).

% sum_diff1_nat
tff(fact_8422_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A4: set_nat,A: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_nat_nat(F,A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) ).

% sum_diff1_nat
tff(fact_8423_powr__realpow,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(powr_real(X3),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(power_power_real(X3),Nb) ) ) ).

% powr_realpow
tff(fact_8424_Iio__Int__singleton,axiom,
    ! [K2: $o,X3: $o] :
      inf_inf_set_o(set_ord_lessThan_o((K2)),insert_o((X3),bot_bot_set_o)) = $ite(aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(K2)),insert_o((X3),bot_bot_set_o),bot_bot_set_o) ).

% Iio_Int_singleton
tff(fact_8425_Iio__Int__singleton,axiom,
    ! [K2: rat,X3: rat] :
      inf_inf_set_rat(set_ord_lessThan_rat(K2),insert_rat(X3,bot_bot_set_rat)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),K2),insert_rat(X3,bot_bot_set_rat),bot_bot_set_rat) ).

% Iio_Int_singleton
tff(fact_8426_Iio__Int__singleton,axiom,
    ! [K2: num,X3: num] :
      inf_inf_set_num(set_ord_lessThan_num(K2),insert_num(X3,bot_bot_set_num)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,X3),K2),insert_num(X3,bot_bot_set_num),bot_bot_set_num) ).

% Iio_Int_singleton
tff(fact_8427_Iio__Int__singleton,axiom,
    ! [K2: int,X3: int] :
      inf_inf_set_int(set_ord_lessThan_int(K2),insert_int(X3,bot_bot_set_int)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),K2),insert_int(X3,bot_bot_set_int),bot_bot_set_int) ).

% Iio_Int_singleton
tff(fact_8428_Iio__Int__singleton,axiom,
    ! [K2: nat,X3: nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2)),insert_nat(X3,bot_bot_set_nat)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),K2),insert_nat(X3,bot_bot_set_nat),bot_bot_set_nat) ).

% Iio_Int_singleton
tff(fact_8429_Iio__Int__singleton,axiom,
    ! [K2: real,X3: real] :
      inf_inf_set_real(set_or5984915006950818249n_real(K2),insert_real(X3,bot_bot_set_real)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X3),K2),insert_real(X3,bot_bot_set_real),bot_bot_set_real) ).

% Iio_Int_singleton
tff(fact_8430_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A4: set_nat,S4: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S2)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( ( S4 = aa_real_real(plus_plus_real(S2),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nh(G),F),A4)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_ni(G),A4),F)),S4) ) ) ) ).

% sums_If_finite_set'
tff(fact_8431_suminf__le__const,axiom,
    ! [F: fun_nat_int,X3: int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),X3) ) ) ).

% suminf_le_const
tff(fact_8432_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X3: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),X3) ) ) ).

% suminf_le_const
tff(fact_8433_suminf__le__const,axiom,
    ! [F: fun_nat_real,X3: real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),X3) ) ) ).

% suminf_le_const
tff(fact_8434_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8435_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8436_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8437_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_8438_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_nj(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_8439_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_nk(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_8440_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_nl(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_8441_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_js(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8442_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_jt(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8443_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_ju(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8444_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X3: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X3)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8445_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X3)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8446_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X3: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X3)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8447_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_8448_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups6591440286371151544t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_8449_sum__Un__nat,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5693394587270226106ex_nat(F,sup_sup_set_complex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4))),groups5693394587270226106ex_nat(F,inf_inf_set_complex(A4,B4))) ) ) ) ).

% sum_Un_nat
tff(fact_8450_sum__Un__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(F,A4)),groups977919841031483927at_nat(F,B4))),groups977919841031483927at_nat(F,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_8451_sum__Un__nat,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,F: fun_Pr1253225528153422666at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
       => ( groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups3860910324918113789at_nat(F,A4)),groups3860910324918113789at_nat(F,B4))),groups3860910324918113789at_nat(F,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_8452_sum__Un__nat,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_8453_power__diff__1__eq,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X3),Nb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X3),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8454_power__diff__1__eq,axiom,
    ! [X3: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X3),Nb)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8455_power__diff__1__eq,axiom,
    ! [X3: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X3),Nb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),one_one_int)),groups3539618377306564664at_int(power_power_int(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8456_power__diff__1__eq,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),Nb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X3),one_one_real)),groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_8457_one__diff__power__eq,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X3),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X3)),groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8458_one__diff__power__eq,axiom,
    ! [X3: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X3),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3)),groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8459_one__diff__power__eq,axiom,
    ! [X3: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X3),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X3)),groups3539618377306564664at_int(power_power_int(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8460_one__diff__power__eq,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X3)),groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_8461_geometric__sum,axiom,
    ! [X3: complex,Nb: nat] :
      ( ( X3 != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X3),Nb)),one_one_complex)),aa_complex_complex(minus_minus_complex(X3),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_8462_geometric__sum,axiom,
    ! [X3: rat,Nb: nat] :
      ( ( X3 != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X3),Nb)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8463_geometric__sum,axiom,
    ! [X3: real,Nb: nat] :
      ( ( X3 != one_one_real )
     => ( groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),Nb)),one_one_real)),aa_real_real(minus_minus_real(X3),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8464_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(plus_plus_real(suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ) ).

% suminf_split_initial_segment
tff(fact_8465_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2)) = aa_real_real(minus_minus_real(suminf_real(F)),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ) ).

% suminf_minus_initial_segment
tff(fact_8466_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_8467_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_8468_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_8469_sum__gp__strict,axiom,
    ! [X3: complex,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(X3 = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X3),Nb))),aa_complex_complex(minus_minus_complex(one_one_complex),X3))) ).

% sum_gp_strict
tff(fact_8470_sum__gp__strict,axiom,
    ! [X3: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(X3 = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X3),Nb))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3))) ).

% sum_gp_strict
tff(fact_8471_sum__gp__strict,axiom,
    ! [X3: real,Nb: nat] :
      groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(X3 = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),Nb))),aa_real_real(minus_minus_real(one_one_real),X3))) ).

% sum_gp_strict
tff(fact_8472_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nm(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8473_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_no(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_np(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8474_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nq(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nr(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8475_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ns(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nt(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_8476_power__diff__sumr2,axiom,
    ! [X3: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X3),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X3),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nu(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8477_power__diff__sumr2,axiom,
    ! [X3: rat,Nb: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X3),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nv(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8478_power__diff__sumr2,axiom,
    ! [X3: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X3),Nb)),aa_nat_int(power_power_int(Y),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nw(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8479_power__diff__sumr2,axiom,
    ! [X3: real,Nb: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),Nb)),aa_nat_real(power_power_real(Y),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X3),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nx(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_8480_diff__power__eq__sum,axiom,
    ! [X3: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X3),aa_nat_nat(suc,Nb))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X3),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ny(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8481_diff__power__eq__sum,axiom,
    ! [X3: rat,Nb: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X3),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nz(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8482_diff__power__eq__sum,axiom,
    ! [X3: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X3),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_oa(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8483_diff__power__eq__sum,axiom,
    ! [X3: real,Nb: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),aa_nat_nat(suc,Nb))),aa_nat_real(power_power_real(Y),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X3),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ob(X3),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_8484_geometric__sums,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => aa_real_o(sums_real(power_power_real(C)),aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C))) ) ).

% geometric_sums
tff(fact_8485_geometric__sums,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => aa_complex_o(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C))) ) ).

% geometric_sums
tff(fact_8486_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_oc),one_one_real) ).

% power_half_series
tff(fact_8487_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K5: rat,K2: nat] :
      ( ! [P3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P3),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P3)),K5) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8488_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K2: nat] :
      ( ! [P3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P3)),K5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8489_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K2: nat] :
      ( ! [P3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P3),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P3)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8490_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K2: nat] :
      ( ! [P3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P3),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P3)),K5) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8491_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8492_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8493_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8494_one__diff__power__eq_H,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X3),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X3)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_od(X3),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8495_one__diff__power__eq_H,axiom,
    ! [X3: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X3),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oe(X3),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8496_one__diff__power__eq_H,axiom,
    ! [X3: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X3),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X3)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_of(X3),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8497_one__diff__power__eq_H,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X3)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_og(X3),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_8498_sums__if_H,axiom,
    ! [G: fun_nat_real,X3: real] :
      ( aa_real_o(sums_real(G),X3)
     => aa_real_o(sums_real(aTP_Lamm_oh(G)),X3) ) ).

% sums_if'
tff(fact_8499_sums__if,axiom,
    ! [G: fun_nat_real,X3: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),X3)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oi(G),F)),aa_real_real(plus_plus_real(X3),Y)) ) ) ).

% sums_if
tff(fact_8500_powr__half__sqrt,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_real(powr_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X3) ) ) ).

% powr_half_sqrt
tff(fact_8501_powr__neg__numeral,axiom,
    ! [X3: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(powr_real(X3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_8502_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_oj(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_ok(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_ol(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_8503_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_om,set_or1266510415728281911st_int(Mb,Nb)) = 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,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_8504_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_8505_mono__SucI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,aa_nat_nat(suc,N)))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI1
tff(fact_8506_mono__SucI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,aa_nat_nat(suc,N)))
     => topolo7278393974255667507et_nat(X6) ) ).

% mono_SucI1
tff(fact_8507_mono__SucI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,aa_nat_nat(suc,N)))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI1
tff(fact_8508_mono__SucI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,aa_nat_nat(suc,N)))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI1
tff(fact_8509_mono__SucI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,aa_nat_nat(suc,N)))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI1
tff(fact_8510_mono__SucI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,aa_nat_nat(suc,N)))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI1
tff(fact_8511_mono__SucI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N))),aa_nat_real(X6,N))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI2
tff(fact_8512_mono__SucI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N))),aa_nat_set_nat(X6,N))
     => topolo7278393974255667507et_nat(X6) ) ).

% mono_SucI2
tff(fact_8513_mono__SucI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N))),aa_nat_rat(X6,N))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI2
tff(fact_8514_mono__SucI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N))),aa_nat_num(X6,N))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI2
tff(fact_8515_mono__SucI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N))),aa_nat_nat(X6,N))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI2
tff(fact_8516_mono__SucI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N))),aa_nat_int(X6,N))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI2
tff(fact_8517_monoseq__Suc,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N3))),aa_nat_real(X6,N3)) ) ) ).

% monoseq_Suc
tff(fact_8518_monoseq__Suc,axiom,
    ! [X6: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X6)
    <=> ( ! [N3: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N3)),aa_nat_set_nat(X6,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N3))),aa_nat_set_nat(X6,N3)) ) ) ).

% monoseq_Suc
tff(fact_8519_monoseq__Suc,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N3))),aa_nat_rat(X6,N3)) ) ) ).

% monoseq_Suc
tff(fact_8520_monoseq__Suc,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N3))),aa_nat_num(X6,N3)) ) ) ).

% monoseq_Suc
tff(fact_8521_monoseq__Suc,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N3))),aa_nat_nat(X6,N3)) ) ) ).

% monoseq_Suc
tff(fact_8522_monoseq__Suc,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N3))),aa_nat_int(X6,N3)) ) ) ).

% monoseq_Suc
tff(fact_8523_monoI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M)),aa_nat_real(X6,N)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI1
tff(fact_8524_monoI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M)),aa_nat_set_nat(X6,N)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI1
tff(fact_8525_monoI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M)),aa_nat_rat(X6,N)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI1
tff(fact_8526_monoI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M)),aa_nat_num(X6,N)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI1
tff(fact_8527_monoI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M)),aa_nat_nat(X6,N)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI1
tff(fact_8528_monoI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M)),aa_nat_int(X6,N)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI1
tff(fact_8529_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8530_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_il(F),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8531_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_on,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8532_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X3: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oo(C),X3))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(C),X3)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_oo(C),X3))) ) ).

% diffs_equiv
tff(fact_8533_diffs__equiv,axiom,
    ! [C: fun_nat_real,X3: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_oq(C),X3))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_or(C),X3)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_oq(C),X3))) ) ).

% diffs_equiv
tff(fact_8534_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S2),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_8535_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [K: nat] : aa_set_nat_o(ord_less_eq_set_nat(S2),aa_nat_set_nat(set_ord_lessThan_nat,K)) ) ).

% finite_nat_bounded
tff(fact_8536_diffs__def,axiom,
    ! [C: fun_nat_rat,X: nat] : aa_nat_rat(diffs_rat(C),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X))),aa_nat_rat(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8537_diffs__def,axiom,
    ! [C: fun_nat_real,X: nat] : aa_nat_real(diffs_real(C),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X))),aa_nat_real(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8538_diffs__def,axiom,
    ! [C: fun_nat_int,X: nat] : aa_nat_int(diffs_int(C),X) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X))),aa_nat_int(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8539_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,X3: complex] :
      ( ! [X5: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),X5))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_oo(C),X3)) ) ).

% termdiff_converges_all
tff(fact_8540_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,X3: real] :
      ( ! [X5: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_io(C),X5))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_oq(C),X3)) ) ).

% termdiff_converges_all
tff(fact_8541_finite__transitivity__chain,axiom,
    ! [A4: set_real,R: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X5: real] : ~ aa_real_o(aa_real_fun_real_o(R,X5),X5)
       => ( ! [X5: real,Y3: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(R,X5),Y3)
             => ( aa_real_o(aa_real_fun_real_o(R,Y3),Z2)
               => aa_real_o(aa_real_fun_real_o(R,X5),Z2) ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),A4)
               => ? [Y5: real] :
                    ( aa_set_real_o(member_real(Y5),A4)
                    & aa_real_o(aa_real_fun_real_o(R,X5),Y5) ) )
           => ( A4 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8542_finite__transitivity__chain,axiom,
    ! [A4: set_complex,R: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X5: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R,X5),X5)
       => ( ! [X5: complex,Y3: complex,Z2: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R,X5),Y3)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R,Y3),Z2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,X5),Z2) ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),A4)
               => ? [Y5: complex] :
                    ( aa_set_complex_o(member_complex(Y5),A4)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R,X5),Y5) ) )
           => ( A4 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8543_finite__transitivity__chain,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R: fun_Pr2028404973800837934_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [X5: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X5),X5)
       => ( ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X5),Y3)
             => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Y3),Z2)
               => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X5),Z2) ) )
         => ( ! [X5: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A4)
               => ? [Y5: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y5),A4)
                    & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X5),Y5) ) )
           => ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8544_finite__transitivity__chain,axiom,
    ! [A4: set_o,R: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X5: $o] : ~ aa_o_o(aa_o_fun_o_o(R,(X5)),(X5))
       => ( ! [X5: $o,Y3: $o,Z2: $o] :
              ( aa_o_o(aa_o_fun_o_o(R,(X5)),(Y3))
             => ( aa_o_o(aa_o_fun_o_o(R,(Y3)),(Z2))
               => aa_o_o(aa_o_fun_o_o(R,(X5)),(Z2)) ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
               => ? [Y5: $o] :
                    ( aa_set_o_o(member_o((Y5)),A4)
                    & aa_o_o(aa_o_fun_o_o(R,(X5)),(Y5)) ) )
           => ( A4 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8545_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X5: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X5),X5)
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X5),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X5),Z2) ) )
         => ( ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
               => ? [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),A4)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X5),Y5) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8546_finite__transitivity__chain,axiom,
    ! [A4: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X5: int] : ~ aa_int_o(aa_int_fun_int_o(R,X5),X5)
       => ( ! [X5: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X5),Y3)
             => ( aa_int_o(aa_int_fun_int_o(R,Y3),Z2)
               => aa_int_o(aa_int_fun_int_o(R,X5),Z2) ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
               => ? [Y5: int] :
                    ( aa_set_int_o(member_int(Y5),A4)
                    & aa_int_o(aa_int_fun_int_o(R,X5),Y5) ) )
           => ( A4 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_8547_termdiff__converges,axiom,
    ! [X3: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),K5)
     => ( ! [X5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X5)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_io(C),X5)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_os(X3),C)) ) ) ).

% termdiff_converges
tff(fact_8548_termdiff__converges,axiom,
    ! [X3: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),K5)
     => ( ! [X5: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X5)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),X5)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ot(X3),C)) ) ) ).

% termdiff_converges
tff(fact_8549_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M4: nat] :
        ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
          & aa_set_nat_o(member_nat(N3),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_8550_Maclaurin__cos__expansion2,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [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),X3)
            & ( aa_real_real(cos_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_ou(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8551_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
            & ( aa_real_real(cos_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_ou(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8552_Maclaurin__cos__expansion,axiom,
    ! [X3: real,Nb: 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,X3))
      & ( aa_real_real(cos_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_ou(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8553_sin__paired,axiom,
    ! [X3: real] : aa_real_o(sums_real(aTP_Lamm_ov(X3)),aa_real_real(sin_real,X3)) ).

% sin_paired
tff(fact_8554_arcosh__def,axiom,
    ! [X3: real] : aa_real_real(arcosh_real,X3) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X3),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_8555_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% of_real_numeral
tff(fact_8556_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_8557_of__real__power,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(power_power_real(X3),Nb)) = aa_nat_real(power_power_real(aa_real_real(real_V1803761363581548252l_real,X3)),Nb) ).

% of_real_power
tff(fact_8558_of__real__power,axiom,
    ! [X3: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(power_power_real(X3),Nb)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,X3)),Nb) ).

% of_real_power
tff(fact_8559_of__real__add,axiom,
    ! [X3: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(plus_plus_real(X3),Y)) = aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X3)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_8560_of__real__add,axiom,
    ! [X3: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(X3),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X3)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_8561_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8562_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8563_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8564_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8565_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8566_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_8567_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_8568_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_8569_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_8570_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8571_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8572_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8573_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8574_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8575_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) ).

% of_real_neg_numeral
tff(fact_8576_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_8577_norm__of__real__add1,axiom,
    ! [X3: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X3)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X3),one_one_real)) ).

% norm_of_real_add1
tff(fact_8578_norm__of__real__add1,axiom,
    ! [X3: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X3)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X3),one_one_real)) ).

% norm_of_real_add1
tff(fact_8579_norm__of__real__addn,axiom,
    ! [X3: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X3)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X3),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_8580_norm__of__real__addn,axiom,
    ! [X3: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X3)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X3),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_8581_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_8582_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_8583_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_8584_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_8585_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_8586_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_8587_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_8588_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_8589_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_8590_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_8591_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_8592_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_8593_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_8594_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_8595_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_8596_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_8597_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_8598_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_8599_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_8600_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_8601_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8602_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : 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,K2)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8603_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : 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,K2)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8604_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8605_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8606_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8607_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Nb),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_8608_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8609_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8610_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8611_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8612_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8613_norm__less__p1,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X3))),one_one_real))) ).

% norm_less_p1
tff(fact_8614_norm__less__p1,axiom,
    ! [X3: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X3))),one_one_complex))) ).

% norm_less_p1
tff(fact_8615_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_8616_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => 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,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_8617_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => 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,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_8618_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_8619_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => 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,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_8620_fact__numeral,axiom,
    ! [K2: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8621_fact__numeral,axiom,
    ! [K2: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8622_fact__numeral,axiom,
    ! [K2: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8623_fact__numeral,axiom,
    ! [K2: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K2)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8624_fact__numeral,axiom,
    ! [K2: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8625_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_8626_cos__sin__eq,axiom,
    ! [X3: real] : aa_real_real(cos_real,X3) = aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)) ).

% cos_sin_eq
tff(fact_8627_cos__sin__eq,axiom,
    ! [X3: complex] : aa_complex_complex(cos_complex,X3) = aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X3)) ).

% cos_sin_eq
tff(fact_8628_sin__cos__eq,axiom,
    ! [X3: real] : aa_real_real(sin_real,X3) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)) ).

% sin_cos_eq
tff(fact_8629_sin__cos__eq,axiom,
    ! [X3: complex] : aa_complex_complex(sin_complex,X3) = aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X3)) ).

% sin_cos_eq
tff(fact_8630_Maclaurin__zero,axiom,
    ! [X3: real,Nb: nat,Diff: fun_na2180513548729900188l_real] :
      ( ( X3 = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun4674562053777771865t_real(aTP_Lamm_ow(X3),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_literal_real(aa_nat8303615512603269253l_real(Diff,zero_zero_nat),zero_zero_literal) ) ) ) ).

% Maclaurin_zero
tff(fact_8631_Maclaurin__zero,axiom,
    ! [X3: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X3 = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ox(X3),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8632_Maclaurin__zero,axiom,
    ! [X3: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X3 = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_oy(X3),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8633_Maclaurin__zero,axiom,
    ! [X3: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X3 = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_oz(X3),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8634_Maclaurin__zero,axiom,
    ! [X3: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( X3 = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_pa(X3),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8635_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B5: real] : aa_real_real(F,H) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pb(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(B5),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_8636_minus__sin__cos__eq,axiom,
    ! [X3: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X3)) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_8637_minus__sin__cos__eq,axiom,
    ! [X3: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X3)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X3),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_8638_Maclaurin__exp__le,axiom,
    ! [X3: real,Nb: 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,X3))
      & ( aa_real_real(exp_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_pc(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_8639_cos__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(cos_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)),zero_zero_real) ).

% cos_coeff_def
tff(fact_8640_cos__paired,axiom,
    ! [X3: real] : aa_real_o(sums_real(aTP_Lamm_pd(X3)),aa_real_real(cos_real,X3)) ).

% cos_paired
tff(fact_8641_arsinh__def,axiom,
    ! [X3: real] : aa_real_real(arsinh_real,X3) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X3),aa_real_real(powr_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_8642_Maclaurin__exp__lt,axiom,
    ! [X3: real,Nb: nat] :
      ( ( X3 != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [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,X3))
            & ( aa_real_real(exp_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_pc(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8643_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
       => ? [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),X3)
            & ( aa_real_real(sin_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_pe(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8644_Maclaurin__sin__expansion4,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ? [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),X3)
          & ( aa_real_real(sin_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_pe(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8645_Maclaurin__sin__expansion2,axiom,
    ! [X3: real,Nb: 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,X3))
      & ( aa_real_real(sin_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_pe(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8646_Maclaurin__sin__expansion,axiom,
    ! [X3: real,Nb: nat] :
    ? [T4: real] : aa_real_real(sin_real,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_pe(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T4),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_8647_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X))) ).

% sin_coeff_def
tff(fact_8648_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_8649_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_8650_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_8651_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_8652_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_8653_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_8654_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_8655_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_8656_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8657_pochhammer__double,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8658_pochhammer__double,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_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_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),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))))),Nb)) ).

% pochhammer_double
tff(fact_8659_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_real(comm_s7457072308508201937r_real(Z),Nb))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8660_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_pf,Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_8661_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_pg,Nb),zero_zero_rat) ).

% of_nat_code
tff(fact_8662_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_ph),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_8663_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_pi,Nb),zero_zero_int) ).

% of_nat_code
tff(fact_8664_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_pj),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_8665_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_pk(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8666_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_pl(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8667_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_pm(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8668_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8669_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_8670_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K2)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_8671_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K2)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_8672_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_8673_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K2)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_8674_floor__numeral,axiom,
    ! [V2: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V2)) = aa_num_int(numeral_numeral_int,V2) ).

% floor_numeral
tff(fact_8675_floor__numeral,axiom,
    ! [V2: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V2)) = aa_num_int(numeral_numeral_int,V2) ).

% floor_numeral
tff(fact_8676_binomial__0__Suc,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8677_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_8678_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_8679_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% zero_less_binomial_iff
tff(fact_8680_zero__le__floor,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3) ) ).

% zero_le_floor
tff(fact_8681_zero__le__floor,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3) ) ).

% zero_le_floor
tff(fact_8682_numeral__le__floor,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V2)),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V2)),X3) ) ).

% numeral_le_floor
tff(fact_8683_numeral__le__floor,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V2)),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,V2)),X3) ) ).

% numeral_le_floor
tff(fact_8684_zero__less__floor,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X3) ) ).

% zero_less_floor
tff(fact_8685_zero__less__floor,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X3) ) ).

% zero_less_floor
tff(fact_8686_floor__less__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_num_rat(numeral_numeral_rat,V2)) ) ).

% floor_less_numeral
tff(fact_8687_floor__less__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_num_real(numeral_numeral_real,V2)) ) ).

% floor_less_numeral
tff(fact_8688_one__le__floor,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X3) ) ).

% one_le_floor
tff(fact_8689_one__le__floor,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X3) ) ).

% one_le_floor
tff(fact_8690_floor__neg__numeral,axiom,
    ! [V2: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)) ).

% floor_neg_numeral
tff(fact_8691_floor__neg__numeral,axiom,
    ! [V2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)) ).

% floor_neg_numeral
tff(fact_8692_floor__diff__numeral,axiom,
    ! [X3: rat,V2: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_num_rat(numeral_numeral_rat,V2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X3)),aa_num_int(numeral_numeral_int,V2)) ).

% floor_diff_numeral
tff(fact_8693_floor__diff__numeral,axiom,
    ! [X3: real,V2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(minus_minus_real(X3),aa_num_real(numeral_numeral_real,V2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X3)),aa_num_int(numeral_numeral_int,V2)) ).

% floor_diff_numeral
tff(fact_8694_floor__numeral__power,axiom,
    ! [X3: num,Nb: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X3)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ).

% floor_numeral_power
tff(fact_8695_floor__numeral__power,axiom,
    ! [X3: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X3)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ).

% floor_numeral_power
tff(fact_8696_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_8697_numeral__less__floor,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V2)),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> 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,V2)),one_one_rat)),X3) ) ).

% numeral_less_floor
tff(fact_8698_numeral__less__floor,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V2)),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V2)),one_one_real)),X3) ) ).

% numeral_less_floor
tff(fact_8699_floor__le__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V2)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_8700_floor__le__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X3)),aa_num_int(numeral_numeral_int,V2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V2)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_8701_one__less__floor,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X3) ) ).

% one_less_floor
tff(fact_8702_one__less__floor,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3) ) ).

% one_less_floor
tff(fact_8703_floor__le__one,axiom,
    ! [X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X3)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8704_floor__le__one,axiom,
    ! [X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X3)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8705_neg__numeral__le__floor,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),X3) ) ).

% neg_numeral_le_floor
tff(fact_8706_neg__numeral__le__floor,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),X3) ) ).

% neg_numeral_le_floor
tff(fact_8707_floor__less__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))) ) ).

% floor_less_neg_numeral
tff(fact_8708_floor__less__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) ) ).

% floor_less_neg_numeral
tff(fact_8709_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8710_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_8711_neg__numeral__less__floor,axiom,
    ! [V2: num,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> 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,V2))),one_one_rat)),X3) ) ).

% neg_numeral_less_floor
tff(fact_8712_neg__numeral__less__floor,axiom,
    ! [V2: num,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),one_one_real)),X3) ) ).

% neg_numeral_less_floor
tff(fact_8713_floor__le__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_8714_floor__le__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X3)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_8715_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_8716_floor__mono,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X3)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ).

% floor_mono
tff(fact_8717_floor__mono,axiom,
    ! [X3: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X3)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_8718_of__int__floor__le,axiom,
    ! [X3: 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,X3))),X3) ).

% of_int_floor_le
tff(fact_8719_of__int__floor__le,axiom,
    ! [X3: 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,X3))),X3) ).

% of_int_floor_le
tff(fact_8720_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))),aa_nat_nat(suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_8721_Suc__times__binomial,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ).

% Suc_times_binomial
tff(fact_8722_binomial__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% binomial_symmetric
tff(fact_8723_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_8724_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_8725_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8726_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8727_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8728_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8729_zero__less__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ).

% zero_less_binomial
tff(fact_8730_le__floor__iff,axiom,
    ! [Z: int,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X3) ) ).

% le_floor_iff
tff(fact_8731_le__floor__iff,axiom,
    ! [Z: int,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X3) ) ).

% le_floor_iff
tff(fact_8732_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_8733_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))),aa_nat_nat(suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8734_le__floor__add,axiom,
    ! [X3: rat,Y: 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,X3)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y))) ).

% le_floor_add
tff(fact_8735_le__floor__add,axiom,
    ! [X3: real,Y: 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,X3)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X3),Y))) ).

% le_floor_add
tff(fact_8736_int__add__floor,axiom,
    ! [Z: int,X3: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X3)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),X3)) ).

% int_add_floor
tff(fact_8737_int__add__floor,axiom,
    ! [Z: int,X3: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X3)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),X3)) ).

% int_add_floor
tff(fact_8738_floor__add__int,axiom,
    ! [X3: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X3)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_8739_floor__add__int,axiom,
    ! [X3: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X3)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X3),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_8740_choose__mult,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ) ).

% choose_mult
tff(fact_8741_binomial__fact__lemma,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8742_gbinomial__pochhammer,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2))),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer
tff(fact_8743_gbinomial__pochhammer,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K2))),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer
tff(fact_8744_gbinomial__pochhammer,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(A),K2) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,A)),K2))),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer
tff(fact_8745_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer'
tff(fact_8746_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer'
tff(fact_8747_gbinomial__pochhammer_H,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(A),K2) = aa_real_real(divide_divide_real(aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer'
tff(fact_8748_floor__power,axiom,
    ! [X3: rat,Nb: nat] :
      ( ( X3 = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X3)) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(X3),Nb)) = aa_nat_int(power_power_int(aa_rat_int(archim3151403230148437115or_rat,X3)),Nb) ) ) ).

% floor_power
tff(fact_8749_floor__power,axiom,
    ! [X3: real,Nb: nat] :
      ( ( X3 = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X3)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(power_power_real(X3),Nb)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,X3)),Nb) ) ) ).

% floor_power
tff(fact_8750_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_8751_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(A),K2)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_8752_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_8753_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),K2) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8754_pochhammer__nonneg,axiom,
    ! [X3: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X3),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8755_pochhammer__nonneg,axiom,
    ! [X3: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X3),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8756_pochhammer__nonneg,axiom,
    ! [X3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X3),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8757_pochhammer__nonneg,axiom,
    ! [X3: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X3),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8758_one__add__floor,axiom,
    ! [X3: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X3)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),one_one_rat)) ).

% one_add_floor
tff(fact_8759_one__add__floor,axiom,
    ! [X3: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X3)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X3),one_one_real)) ).

% one_add_floor
tff(fact_8760_binomial__absorption,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ).

% binomial_absorption
tff(fact_8761_binomial__altdef__nat,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_8762_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_addition_formula
tff(fact_8763_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K2)) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K2))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_addition_formula
tff(fact_8764_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = 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(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ).

% gbinomial_addition_formula
tff(fact_8765_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K2)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(gbinomial_real(A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8766_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),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,K2))),K2)),aa_nat_rat(gbinomial_rat(A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8767_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_8768_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] : aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K2)),A) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(gbinomial_real(A),K2))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_8769_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_8770_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] : aa_real_real(times_times_real(A),aa_nat_real(gbinomial_real(A),K2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(gbinomial_real(A),K2))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_8771_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_8772_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_8773_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_8774_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_8775_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_8776_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_8777_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(A),Nb)),aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_8778_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_8779_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_8780_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8781_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(comm_s7457072308508201937r_real(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8782_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8783_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8784_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8785_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8786_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8787_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8788_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8789_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8790_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z),Nb)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8791_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8792_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8793_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_8794_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(plus_plus_real(aa_int_real(ring_1_of_int_real,I4)),one_one_real)) )
         => aa_int_o(P,I4) ) ) ).

% floor_split
tff(fact_8795_floor__eq__iff,axiom,
    ! [X3: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X3) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),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_8796_floor__eq__iff,axiom,
    ! [X3: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X3) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X3)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_8797_floor__unique,axiom,
    ! [Z: int,X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),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,X3) = Z ) ) ) ).

% floor_unique
tff(fact_8798_floor__unique,axiom,
    ! [Z: int,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X3) = Z ) ) ) ).

% floor_unique
tff(fact_8799_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8800_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => 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,Nb)),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8801_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% le_mult_floor
tff(fact_8802_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B))) ) ) ).

% le_mult_floor
tff(fact_8803_less__floor__iff,axiom,
    ! [Z: int,X3: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X3))
    <=> 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)),X3) ) ).

% less_floor_iff
tff(fact_8804_less__floor__iff,axiom,
    ! [Z: int,X3: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X3))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X3) ) ).

% less_floor_iff
tff(fact_8805_binomial__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( 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))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_8806_binomial__maximum_H,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8807_floor__le__iff,axiom,
    ! [X3: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X3)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),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_8808_floor__le__iff,axiom,
    ! [X3: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X3)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_8809_binomial__maximum,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8810_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_antimono
tff(fact_8811_binomial__le__pow2,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8812_floor__correct,axiom,
    ! [X3: 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,X3))),X3)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X3)),one_one_int))) ) ).

% floor_correct
tff(fact_8813_floor__correct,axiom,
    ! [X3: 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,X3))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X3)),one_one_int))) ) ).

% floor_correct
tff(fact_8814_choose__reduce__nat,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_8815_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K2)) ).

% Suc_times_gbinomial
tff(fact_8816_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K2)) ).

% Suc_times_gbinomial
tff(fact_8817_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(gbinomial_real(A),K2)) ).

% Suc_times_gbinomial
tff(fact_8818_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorption
tff(fact_8819_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ).

% gbinomial_absorption
tff(fact_8820_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K2))) = aa_real_real(times_times_real(A),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_absorption
tff(fact_8821_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8822_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),Mb)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),K2)) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K2)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8823_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8824_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z),Nb) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z),Mb)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8825_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8826_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8827_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,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,P2),Q3)))),Q3)),P2) ) ).

% floor_divide_lower
tff(fact_8828_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P2),Q3)))),Q3)),P2) ) ).

% floor_divide_lower
tff(fact_8829_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ) ).

% binomial_less_binomial_Suc
tff(fact_8830_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( 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))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8831_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8832_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8833_binomial__addition__formula,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_8834_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8835_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8836_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8837_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8838_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8839_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8840_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))),aa_nat_complex(gbinomial_complex(A),K2)) ).

% gbinomial_factors
tff(fact_8841_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))),aa_nat_rat(gbinomial_rat(A),K2)) ).

% gbinomial_factors
tff(fact_8842_gbinomial__factors,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))),aa_nat_real(gbinomial_real(A),K2)) ).

% gbinomial_factors
tff(fact_8843_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_8844_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_8845_gbinomial__rec,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K2)),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_8846_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),Nb)) ).

% gbinomial_index_swap
tff(fact_8847_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),Nb)) ).

% gbinomial_index_swap
tff(fact_8848_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),K2)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),Nb)) ).

% gbinomial_index_swap
tff(fact_8849_gbinomial__negated__upper,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K2)),A)),one_one_complex)),K2)) ).

% gbinomial_negated_upper
tff(fact_8850_gbinomial__negated__upper,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A)),one_one_rat)),K2)) ).

% gbinomial_negated_upper
tff(fact_8851_gbinomial__negated__upper,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(A),K2) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,K2)),A)),one_one_real)),K2)) ).

% gbinomial_negated_upper
tff(fact_8852_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K2)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8853_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8854_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8855_pochhammer__absorb__comp,axiom,
    ! [R2: real,K2: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K2)) = aa_real_real(times_times_real(R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,R2)),one_one_real)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8856_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K2))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8857_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),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,P2),Q3)))),one_one_rat)),Q3)) ) ).

% floor_divide_upper
tff(fact_8858_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P2),Q3)))),one_one_real)),Q3)) ) ).

% floor_divide_upper
tff(fact_8859_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% pochhammer_same
tff(fact_8860_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% pochhammer_same
tff(fact_8861_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_8862_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% pochhammer_same
tff(fact_8863_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% pochhammer_same
tff(fact_8864_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8865_round__def,axiom,
    ! [X3: rat] : aa_rat_int(archim7778729529865785530nd_rat,X3) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),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))))) ).

% round_def
tff(fact_8866_round__def,axiom,
    ! [X3: real] : aa_real_int(archim8280529875227126926d_real,X3) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8867_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% gbinomial_minus
tff(fact_8868_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% gbinomial_minus
tff(fact_8869_gbinomial__minus,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K2) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% gbinomial_minus
tff(fact_8870_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(gbinomial_complex(A),K2) = 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,K2),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8871_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(gbinomial_real(A),K2) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8872_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = 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,K2),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8873_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2)) ).

% pochhammer_minus'
tff(fact_8874_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2)) ).

% pochhammer_minus'
tff(fact_8875_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2)) ).

% pochhammer_minus'
tff(fact_8876_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K2)) ).

% pochhammer_minus'
tff(fact_8877_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_8878_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% pochhammer_minus
tff(fact_8879_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2)) ).

% pochhammer_minus
tff(fact_8880_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_8881_pochhammer__minus,axiom,
    ! [B: real,K2: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K2) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% pochhammer_minus
tff(fact_8882_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_8883_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_pn(K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8884_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_po(K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8885_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pp,K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8886_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8887_fact__double,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_8888_fact__double,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_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)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_8889_fact__double,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_8890_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8891_binomial__code,axiom,
    ! [Nb: nat,K2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2))) ) ).

% binomial_code
tff(fact_8892_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups6464643781859351333omplex(aTP_Lamm_pq(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8893_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),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_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_pr(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8894_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,Nb))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups129246275422532515t_real(aTP_Lamm_ps(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8895_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_pt(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_8896_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_pu(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_8897_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_pv(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8898_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_pw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8899_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_px(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8900_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_pk(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8901_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_pl(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8902_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_pm(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8903_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_py(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8904_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_pz(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8905_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_qa,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8906_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qb,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8907_atMost__iff,axiom,
    ! [I: real,K2: real] :
      ( aa_set_real_o(member_real(I),set_ord_atMost_real(K2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K2) ) ).

% atMost_iff
tff(fact_8908_atMost__iff,axiom,
    ! [I: $o,K2: $o] :
      ( aa_set_o_o(member_o((I)),set_ord_atMost_o((K2)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(I)),(K2)) ) ).

% atMost_iff
tff(fact_8909_atMost__iff,axiom,
    ! [I: set_nat,K2: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(I),K2) ) ).

% atMost_iff
tff(fact_8910_atMost__iff,axiom,
    ! [I: rat,K2: rat] :
      ( aa_set_rat_o(member_rat(I),set_ord_atMost_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K2) ) ).

% atMost_iff
tff(fact_8911_atMost__iff,axiom,
    ! [I: num,K2: num] :
      ( aa_set_num_o(member_num(I),set_ord_atMost_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),K2) ) ).

% atMost_iff
tff(fact_8912_atMost__iff,axiom,
    ! [I: int,K2: int] :
      ( aa_set_int_o(member_int(I),set_ord_atMost_int(K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K2) ) ).

% atMost_iff
tff(fact_8913_atMost__iff,axiom,
    ! [I: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2) ) ).

% atMost_iff
tff(fact_8914_prod_Oempty,axiom,
    ! [G: fun_o_complex] : groups4859619685533338977omplex(G,bot_bot_set_o) = one_one_complex ).

% prod.empty
tff(fact_8915_prod_Oempty,axiom,
    ! [G: fun_o_real] : groups234877984723959775o_real(G,bot_bot_set_o) = one_one_real ).

% prod.empty
tff(fact_8916_prod_Oempty,axiom,
    ! [G: fun_o_rat] : groups2869687844427037835_o_rat(G,bot_bot_set_o) = one_one_rat ).

% prod.empty
tff(fact_8917_prod_Oempty,axiom,
    ! [G: fun_o_nat] : groups3504817904513533571_o_nat(G,bot_bot_set_o) = one_one_nat ).

% prod.empty
tff(fact_8918_prod_Oempty,axiom,
    ! [G: fun_o_int] : groups3502327434004483295_o_int(G,bot_bot_set_o) = one_one_int ).

% prod.empty
tff(fact_8919_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_8920_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8921_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8922_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_8923_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_8924_atMost__subset__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X3),set_or4236626031148496127et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(X3),Y) ) ).

% atMost_subset_iff
tff(fact_8925_atMost__subset__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_atMost_rat(X3)),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y) ) ).

% atMost_subset_iff
tff(fact_8926_atMost__subset__iff,axiom,
    ! [X3: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_atMost_num(X3)),set_ord_atMost_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y) ) ).

% atMost_subset_iff
tff(fact_8927_atMost__subset__iff,axiom,
    ! [X3: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(X3)),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y) ) ).

% atMost_subset_iff
tff(fact_8928_atMost__subset__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X3)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ).

% atMost_subset_iff
tff(fact_8929_prod_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(X3),A4)
       => ( groups1681761925125756287l_real(G,insert_real(X3,A4)) = aa_real_real(times_times_real(aa_real_real(G,X3)),groups1681761925125756287l_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8930_prod_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X3)),A4)
       => ( groups234877984723959775o_real(G,insert_o((X3),A4)) = aa_real_real(times_times_real(aa_o_real(G,(X3))),groups234877984723959775o_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8931_prod_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(X3),A4)
       => ( groups2316167850115554303t_real(G,insert_int(X3,A4)) = aa_real_real(times_times_real(aa_int_real(G,X3)),groups2316167850115554303t_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8932_prod_Oinsert,axiom,
    ! [A4: set_nat,X3: nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(X3),A4)
       => ( groups129246275422532515t_real(G,insert_nat(X3,A4)) = aa_real_real(times_times_real(aa_nat_real(G,X3)),groups129246275422532515t_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8933_prod_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(X3),A4)
       => ( groups766887009212190081x_real(G,insert_complex(X3,A4)) = aa_real_real(times_times_real(aa_complex_real(G,X3)),groups766887009212190081x_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8934_prod_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(X3),A4)
       => ( groups4061424788464935467al_rat(G,insert_real(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G,X3)),groups4061424788464935467al_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8935_prod_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X3)),A4)
       => ( groups2869687844427037835_o_rat(G,insert_o((X3),A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X3))),groups2869687844427037835_o_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8936_prod_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(X3),A4)
       => ( groups1072433553688619179nt_rat(G,insert_int(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X3)),groups1072433553688619179nt_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8937_prod_Oinsert,axiom,
    ! [A4: set_nat,X3: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(X3),A4)
       => ( groups73079841787564623at_rat(G,insert_nat(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X3)),groups73079841787564623at_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8938_prod_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(X3),A4)
       => ( groups225925009352817453ex_rat(G,insert_complex(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G,X3)),groups225925009352817453ex_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_8939_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H),set_or4236626031148496127et_nat(H2))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
        | aa_set_nat_o(ord_less_eq_set_nat(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8940_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(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,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8941_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(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,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8942_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,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,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8943_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(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,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8944_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(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,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8945_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8946_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8947_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8948_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8949_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8950_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8951_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8952_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8953_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8954_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8955_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8956_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8957_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_8958_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8959_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8960_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8961_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8962_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8963_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: $o] : bot_bot_set_o != set_ord_atMost_o((H)) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8964_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8965_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8966_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qc(F),Nb)),A4) ).

% prod_power_distrib
tff(fact_8967_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A4: set_nat,Nb: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A4)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qd(F),Nb),A4) ).

% prod_power_distrib
tff(fact_8968_prod__power__distrib,axiom,
    ! [F: fun_int_int,A4: set_int,Nb: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A4)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_qe(F),Nb),A4) ).

% prod_power_distrib
tff(fact_8969_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8970_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8971_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8972_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_8973_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qf(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qh(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_8974_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,Nb: nat] : groups705719431365010083at_int(aTP_Lamm_qi(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qk(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_8975_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_et(U)) ).

% atMost_def
tff(fact_8976_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_ql,U)) ).

% atMost_def
tff(fact_8977_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_qm,U)) ).

% atMost_def
tff(fact_8978_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_qn,U)) ).

% atMost_def
tff(fact_8979_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eq,U)) ).

% atMost_def
tff(fact_8980_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X5)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) ) ).

% prod_nonneg
tff(fact_8981_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X5)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A4)) ) ).

% prod_nonneg
tff(fact_8982_prod__nonneg,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X5)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A4)) ) ).

% prod_nonneg
tff(fact_8983_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I3)),aa_complex_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4)) ) ).

% prod_mono
tff(fact_8984_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I3)),aa_real_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4)) ) ).

% prod_mono
tff(fact_8985_prod__mono,axiom,
    ! [A4: set_o,F: fun_o_real,G: fun_o_real] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I3)))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I3))),aa_o_real(G,(I3))) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups234877984723959775o_real(F,A4)),groups234877984723959775o_real(G,A4)) ) ).

% prod_mono
tff(fact_8986_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A4)),groups129246275422532515t_real(G,A4)) ) ).

% prod_mono
tff(fact_8987_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I3)),aa_int_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4)) ) ).

% prod_mono
tff(fact_8988_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4)) ) ).

% prod_mono
tff(fact_8989_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4)) ) ).

% prod_mono
tff(fact_8990_prod__mono,axiom,
    ! [A4: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3)))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A4)),groups2869687844427037835_o_rat(G,A4)) ) ).

% prod_mono
tff(fact_8991_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4)) ) ).

% prod_mono
tff(fact_8992_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4)) ) ).

% prod_mono
tff(fact_8993_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A4)) ) ).

% prod_ge_1
tff(fact_8994_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A4)) ) ).

% prod_ge_1
tff(fact_8995_prod__ge__1,axiom,
    ! [A4: set_o,F: fun_o_real] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_o_real(F,(X5))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups234877984723959775o_real(F,A4)) ) ).

% prod_ge_1
tff(fact_8996_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A4)) ) ).

% prod_ge_1
tff(fact_8997_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X5)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A4)) ) ).

% prod_ge_1
tff(fact_8998_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_8999_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_9000_prod__ge__1,axiom,
    ! [A4: set_o,F: fun_o_rat] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(X5))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups2869687844427037835_o_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_9001_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_9002_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X5)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_9003_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9004_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9005_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9006_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9007_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_ord_atMost_nat,K2) ).

% lessThan_Suc_atMost
tff(fact_9008_atMost__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2)) = insert_nat(aa_nat_nat(suc,K2),aa_nat_set_nat(set_ord_atMost_nat,K2)) ).

% atMost_Suc
tff(fact_9009_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(H)),set_or1266510415728281911st_int(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_9010_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ aa_set_real_o(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_9011_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_9012_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_9013_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A4: set_nat] : aa_nat_real(power_power_real(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_qo(C),F),A4) ).

% power_sum
tff(fact_9014_power__sum,axiom,
    ! [C: complex,F: fun_nat_nat,A4: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_qp(C),F),A4) ).

% power_sum
tff(fact_9015_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qq(C),F)),A4) ).

% power_sum
tff(fact_9016_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A4: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_qr(C),F),A4) ).

% power_sum
tff(fact_9017_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A4: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A4)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_qs(C),F),A4) ).

% power_sum
tff(fact_9018_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K2: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_it(G),K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_9019_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K2: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qt(G),K2),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_9020_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S2),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_9021_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X5)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_9022_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X5)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_9023_prod__le__1,axiom,
    ! [A4: set_o,F: fun_o_real] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X5)))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(X5))),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups234877984723959775o_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_9024_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X5))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X5)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_9025_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X5)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_9026_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X5)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_9027_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X5)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_9028_prod__le__1,axiom,
    ! [A4: set_o,F: fun_o_rat] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X5)))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(X5))),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_9029_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X5)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_9030_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X5)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_9031_prod_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_real,X3: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups1681761925125756287l_real(G,insert_real(X3,A4)) = $ite(aa_set_real_o(member_real(X3),A4),groups1681761925125756287l_real(G,A4),aa_real_real(times_times_real(aa_real_real(G,X3)),groups1681761925125756287l_real(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9032_prod_Oinsert__if,axiom,
    ! [A4: set_o,G: fun_o_real,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups234877984723959775o_real(G,insert_o((X3),A4)) = $ite(aa_set_o_o(member_o((X3)),A4),groups234877984723959775o_real(G,A4),aa_real_real(times_times_real(aa_o_real(G,(X3))),groups234877984723959775o_real(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9033_prod_Oinsert__if,axiom,
    ! [A4: set_int,G: fun_int_real,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups2316167850115554303t_real(G,insert_int(X3,A4)) = $ite(aa_set_int_o(member_int(X3),A4),groups2316167850115554303t_real(G,A4),aa_real_real(times_times_real(aa_int_real(G,X3)),groups2316167850115554303t_real(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9034_prod_Oinsert__if,axiom,
    ! [A4: set_nat,G: fun_nat_real,X3: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups129246275422532515t_real(G,insert_nat(X3,A4)) = $ite(aa_set_nat_o(member_nat(X3),A4),groups129246275422532515t_real(G,A4),aa_real_real(times_times_real(aa_nat_real(G,X3)),groups129246275422532515t_real(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9035_prod_Oinsert__if,axiom,
    ! [A4: set_complex,G: fun_complex_real,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups766887009212190081x_real(G,insert_complex(X3,A4)) = $ite(aa_set_complex_o(member_complex(X3),A4),groups766887009212190081x_real(G,A4),aa_real_real(times_times_real(aa_complex_real(G,X3)),groups766887009212190081x_real(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9036_prod_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_rat,X3: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups4061424788464935467al_rat(G,insert_real(X3,A4)) = $ite(aa_set_real_o(member_real(X3),A4),groups4061424788464935467al_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G,X3)),groups4061424788464935467al_rat(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9037_prod_Oinsert__if,axiom,
    ! [A4: set_o,G: fun_o_rat,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups2869687844427037835_o_rat(G,insert_o((X3),A4)) = $ite(aa_set_o_o(member_o((X3)),A4),groups2869687844427037835_o_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X3))),groups2869687844427037835_o_rat(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9038_prod_Oinsert__if,axiom,
    ! [A4: set_int,G: fun_int_rat,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(G,insert_int(X3,A4)) = $ite(aa_set_int_o(member_int(X3),A4),groups1072433553688619179nt_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X3)),groups1072433553688619179nt_rat(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9039_prod_Oinsert__if,axiom,
    ! [A4: set_nat,G: fun_nat_rat,X3: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups73079841787564623at_rat(G,insert_nat(X3,A4)) = $ite(aa_set_nat_o(member_nat(X3),A4),groups73079841787564623at_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X3)),groups73079841787564623at_rat(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9040_prod_Oinsert__if,axiom,
    ! [A4: set_complex,G: fun_complex_rat,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups225925009352817453ex_rat(G,insert_complex(X3,A4)) = $ite(aa_set_complex_o(member_complex(X3),A4),groups225925009352817453ex_rat(G,A4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G,X3)),groups225925009352817453ex_rat(G,A4))) ) ) ).

% prod.insert_if
tff(fact_9041_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9042_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_o_nat(F,(A3))),aa_o_nat(G,(A3))) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3504817904513533571_o_nat(F,A4)),groups3504817904513533571_o_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9043_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A3)),aa_int_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9044_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9045_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9046_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_o_int(F,(A3))),aa_o_int(G,(A3))) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3502327434004483295_o_int(F,A4)),groups3502327434004483295_o_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9047_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9048_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_re6978948838062957830nteger,G: fun_re6978948838062957830nteger] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_real_Code_integer(F,A3)),aa_real_Code_integer(G,A3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups6225526099057966256nteger(F,A4)),groups6225526099057966256nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9049_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_o_Code_integer(F,(A3))),aa_o_Code_integer(G,(A3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups7694694392188491536nteger(F,A4)),groups7694694392188491536nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9050_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,A3)),aa_int_Code_integer(G,A3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A4)),groups3827104343326376752nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_9051_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9052_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4077766827762148844at_nat(F,A4)),groups4077766827762148844at_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9053_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9054_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4075276357253098568at_int(F,A4)),groups4075276357253098568at_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9055_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A4)),groups8682486955453173170nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9056_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups1230400874837758585nteger(F,A4)),groups1230400874837758585nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9057_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A4)),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9058_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)),aa_set_nat_nat(groups708209901874060359at_nat(F),B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9059_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups705719431365010083at_int(F,A4)),groups705719431365010083at_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9060_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A4)),groups1705073143266064639nt_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_9061_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_qu(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9062_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_qv(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9063_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qw(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9064_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_qx(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9065_atMost__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(numeral_numeral_nat,K2),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K2))) ).

% atMost_nat_numeral
tff(fact_9066_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_9067_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qy(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_9068_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_atMost_rat(A)),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_9069_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_atMost_num(A)),set_ord_lessThan_num(B))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_9070_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_9071_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_9072_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_ord_atMost_real(A)),set_or5984915006950818249n_real(B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_9073_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_je(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_9074_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_qz(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_9075_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ra(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rb(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_9076_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rc(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rd(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_9077_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_re(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rf(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_9078_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rg(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rh(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_9079_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ri(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rj(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_9080_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9081_less__1__prod2,axiom,
    ! [I5: set_o,I: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I)),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_o_real(F,(I)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_o_real(F,(I3))) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups234877984723959775o_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9082_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9083_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups129246275422532515t_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9084_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9085_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9086_less__1__prod2,axiom,
    ! [I5: set_o,I: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I)),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(F,(I)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9087_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9088_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9089_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_9090_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rk,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_9091_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9092_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9093_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_o_real(F,(I3))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups234877984723959775o_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9094_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups129246275422532515t_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9095_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9096_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9097_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9098_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(F,(I3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9099_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9100_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_9101_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups766887009212190081x_real(G,A4) = aa_real_real(times_times_real(groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,B4))),groups766887009212190081x_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9102_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups225925009352817453ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups225925009352817453ex_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9103_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups861055069439313189ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9104_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups858564598930262913ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A4,B4))),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9105_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups129246275422532515t_real(G,A4) = aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups129246275422532515t_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9106_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups73079841787564623at_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9107_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups708209901874060359at_nat(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9108_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups705719431365010083at_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups705719431365010083at_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9109_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups1705073143266064639nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,minus_minus_set_int(A4,B4))),groups1705073143266064639nt_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9110_prod_Osubset__diff,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,G: fun_Pr672294278902479040t_real] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ( groups6036352826371341000t_real(G,A4) = aa_real_real(times_times_real(groups6036352826371341000t_real(G,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))),groups6036352826371341000t_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_9111_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_complex(G,X5) = one_one_complex ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_complex(G,X5) = aa_real_complex(H,X5) ) )
           => ( groups713298508707869441omplex(G,T2) = groups713298508707869441omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9112_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S2: set_o,G: fun_o_complex,H: fun_o_complex] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_complex(G,(X5)) = one_one_complex ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_complex(G,(X5)) = aa_o_complex(H,(X5)) ) )
           => ( groups4859619685533338977omplex(G,T2) = groups4859619685533338977omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9113_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(G,X5) = one_one_complex ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_complex(G,X5) = aa_int_complex(H,X5) ) )
           => ( groups7440179247065528705omplex(G,T2) = groups7440179247065528705omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9114_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_complex(G,X5) = one_one_complex ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_complex(G,X5) = aa_complex_complex(H,X5) ) )
           => ( groups3708469109370488835omplex(G,T2) = groups3708469109370488835omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9115_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_real(G,X5) = one_one_real ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_real(G,X5) = aa_real_real(H,X5) ) )
           => ( groups1681761925125756287l_real(G,T2) = groups1681761925125756287l_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9116_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S2: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_real(G,(X5)) = one_one_real ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_real(G,(X5)) = aa_o_real(H,(X5)) ) )
           => ( groups234877984723959775o_real(G,T2) = groups234877984723959775o_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9117_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_real(G,X5) = one_one_real ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_real(G,X5) = aa_int_real(H,X5) ) )
           => ( groups2316167850115554303t_real(G,T2) = groups2316167850115554303t_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9118_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X5) = one_one_real ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_real(G,X5) = aa_complex_real(H,X5) ) )
           => ( groups766887009212190081x_real(G,T2) = groups766887009212190081x_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9119_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(G,X5) = one_one_rat ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_rat(G,X5) = aa_real_rat(H,X5) ) )
           => ( groups4061424788464935467al_rat(G,T2) = groups4061424788464935467al_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9120_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S2: set_o,G: fun_o_rat,H: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_rat(G,(X5)) = one_one_rat ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_rat(G,(X5)) = aa_o_rat(H,(X5)) ) )
           => ( groups2869687844427037835_o_rat(G,T2) = groups2869687844427037835_o_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_9121_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_complex(H,X5) = one_one_complex ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_complex(G,X5) = aa_real_complex(H,X5) ) )
           => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9122_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S2: set_o,H: fun_o_complex,G: fun_o_complex] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_complex(H,(X5)) = one_one_complex ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_complex(G,(X5)) = aa_o_complex(H,(X5)) ) )
           => ( groups4859619685533338977omplex(G,S2) = groups4859619685533338977omplex(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9123_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(H,X5) = one_one_complex ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_complex(G,X5) = aa_int_complex(H,X5) ) )
           => ( groups7440179247065528705omplex(G,S2) = groups7440179247065528705omplex(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9124_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_complex(H,X5) = one_one_complex ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_complex(G,X5) = aa_complex_complex(H,X5) ) )
           => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9125_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_real(H,X5) = one_one_real ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_real(G,X5) = aa_real_real(H,X5) ) )
           => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9126_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S2: set_o,H: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_real(H,(X5)) = one_one_real ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_real(G,(X5)) = aa_o_real(H,(X5)) ) )
           => ( groups234877984723959775o_real(G,S2) = groups234877984723959775o_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9127_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X5: int] :
              ( aa_set_int_o(member_int(X5),minus_minus_set_int(T2,S2))
             => ( aa_int_real(H,X5) = one_one_real ) )
         => ( ! [X5: int] :
                ( aa_set_int_o(member_int(X5),S2)
               => ( aa_int_real(G,X5) = aa_int_real(H,X5) ) )
           => ( groups2316167850115554303t_real(G,S2) = groups2316167850115554303t_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9128_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(H,X5) = one_one_real ) )
         => ( ! [X5: complex] :
                ( aa_set_complex_o(member_complex(X5),S2)
               => ( aa_complex_real(G,X5) = aa_complex_real(H,X5) ) )
           => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9129_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X5: real] :
              ( aa_set_real_o(member_real(X5),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(H,X5) = one_one_rat ) )
         => ( ! [X5: real] :
                ( aa_set_real_o(member_real(X5),S2)
               => ( aa_real_rat(G,X5) = aa_real_rat(H,X5) ) )
           => ( groups4061424788464935467al_rat(G,S2) = groups4061424788464935467al_rat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9130_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S2: set_o,H: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S2),T2)
       => ( ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),minus_minus_set_o(T2,S2))
             => ( aa_o_rat(H,(X5)) = one_one_rat ) )
         => ( ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),S2)
               => ( aa_o_rat(G,(X5)) = aa_o_rat(H,(X5)) ) )
           => ( groups2869687844427037835_o_rat(G,S2) = groups2869687844427037835_o_rat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_9131_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_complex(G,X5) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T2) = groups3708469109370488835omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9132_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X5) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T2) = groups766887009212190081x_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9133_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X5) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T2) = groups225925009352817453ex_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9134_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X5) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T2) = groups861055069439313189ex_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9135_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X5) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T2) = groups858564598930262913ex_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9136_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_complex(G,X5) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,T2) = groups6464643781859351333omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9137_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_real(G,X5) = one_one_real ) )
         => ( groups129246275422532515t_real(G,T2) = groups129246275422532515t_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9138_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_rat(G,X5) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,T2) = groups73079841787564623at_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9139_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_nat(G,X5) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),T2) = aa_set_nat_nat(groups708209901874060359at_nat(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9140_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_int(G,X5) = one_one_int ) )
         => ( groups705719431365010083at_int(G,T2) = groups705719431365010083at_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_9141_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_complex(G,X5) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9142_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X5) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9143_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X5) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9144_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X5) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9145_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X5) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9146_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_complex(G,X5) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,S2) = groups6464643781859351333omplex(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9147_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_real(G,X5) = one_one_real ) )
         => ( groups129246275422532515t_real(G,S2) = groups129246275422532515t_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9148_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_rat(G,X5) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,S2) = groups73079841787564623at_rat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9149_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_nat(G,X5) = one_one_nat ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),S2) = aa_set_nat_nat(groups708209901874060359at_nat(G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9150_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S2))
             => ( aa_nat_int(G,X5) = one_one_int ) )
         => ( groups705719431365010083at_int(G,S2) = groups705719431365010083at_int(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_9151_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C2) = groups713298508707869441omplex(H,C2) )
               => ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9152_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_complex,H: fun_o_complex] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_complex(G,(A3)) = one_one_complex ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_complex(H,(B3)) = one_one_complex ) )
             => ( ( groups4859619685533338977omplex(G,C2) = groups4859619685533338977omplex(H,C2) )
               => ( groups4859619685533338977omplex(G,A4) = groups4859619685533338977omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9153_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_complex(H,B3) = one_one_complex ) )
             => ( ( groups7440179247065528705omplex(G,C2) = groups7440179247065528705omplex(H,C2) )
               => ( groups7440179247065528705omplex(G,A4) = groups7440179247065528705omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9154_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C2) = groups3708469109370488835omplex(H,C2) )
               => ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9155_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C2) = groups1681761925125756287l_real(H,C2) )
               => ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9156_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_real(G,(A3)) = one_one_real ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_real(H,(B3)) = one_one_real ) )
             => ( ( groups234877984723959775o_real(G,C2) = groups234877984723959775o_real(H,C2) )
               => ( groups234877984723959775o_real(G,A4) = groups234877984723959775o_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9157_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_real(G,A3) = one_one_real ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_real(H,B3) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,C2) = groups2316167850115554303t_real(H,C2) )
               => ( groups2316167850115554303t_real(G,A4) = groups2316167850115554303t_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9158_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C2) = groups766887009212190081x_real(H,C2) )
               => ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9159_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C2) = groups4061424788464935467al_rat(H,C2) )
               => ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9160_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_rat,H: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_rat(G,(A3)) = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_rat(H,(B3)) = one_one_rat ) )
             => ( ( groups2869687844427037835_o_rat(G,C2) = groups2869687844427037835_o_rat(H,C2) )
               => ( groups2869687844427037835_o_rat(G,A4) = groups2869687844427037835_o_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_9161_prod_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) )
              <=> ( groups713298508707869441omplex(G,C2) = groups713298508707869441omplex(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9162_prod_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_complex,H: fun_o_complex] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_complex(G,(A3)) = one_one_complex ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_complex(H,(B3)) = one_one_complex ) )
             => ( ( groups4859619685533338977omplex(G,A4) = groups4859619685533338977omplex(H,B4) )
              <=> ( groups4859619685533338977omplex(G,C2) = groups4859619685533338977omplex(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9163_prod_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_complex(H,B3) = one_one_complex ) )
             => ( ( groups7440179247065528705omplex(G,A4) = groups7440179247065528705omplex(H,B4) )
              <=> ( groups7440179247065528705omplex(G,C2) = groups7440179247065528705omplex(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9164_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) )
              <=> ( groups3708469109370488835omplex(G,C2) = groups3708469109370488835omplex(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9165_prod_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) )
              <=> ( groups1681761925125756287l_real(G,C2) = groups1681761925125756287l_real(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9166_prod_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_real(G,(A3)) = one_one_real ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_real(H,(B3)) = one_one_real ) )
             => ( ( groups234877984723959775o_real(G,A4) = groups234877984723959775o_real(H,B4) )
              <=> ( groups234877984723959775o_real(G,C2) = groups234877984723959775o_real(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9167_prod_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C2)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C2)
       => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C2,A4))
               => ( aa_int_real(G,A3) = one_one_real ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(C2,B4))
                 => ( aa_int_real(H,B3) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,A4) = groups2316167850115554303t_real(H,B4) )
              <=> ( groups2316167850115554303t_real(G,C2) = groups2316167850115554303t_real(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9168_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C2,A4))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) )
              <=> ( groups766887009212190081x_real(G,C2) = groups766887009212190081x_real(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9169_prod_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C2)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C2)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C2)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C2,A4))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) )
              <=> ( groups4061424788464935467al_rat(G,C2) = groups4061424788464935467al_rat(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9170_prod_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: fun_o_rat,H: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C2)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C2)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C2)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C2,A4))
               => ( aa_o_rat(G,(A3)) = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C2,B4))
                 => ( aa_o_rat(H,(B3)) = one_one_rat ) )
             => ( ( groups2869687844427037835_o_rat(G,A4) = groups2869687844427037835_o_rat(H,B4) )
              <=> ( groups2869687844427037835_o_rat(G,C2) = groups2869687844427037835_o_rat(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_9171_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9172_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9173_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9174_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9175_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9176_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9177_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9178_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9179_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9180_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,Mb)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9181_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9182_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9183_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_9184_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_9185_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_9186_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_9187_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),groups129246275422532515t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9188_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,Mb)),groups73079841787564623at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9189_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9190_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9191_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9192_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9193_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9194_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9195_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_nj(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_9196_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_nk(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_9197_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : groups6591440286371151544t_real(aTP_Lamm_nl(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_9198_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_9199_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups705719431365010083at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_9200_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(D),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_complex(C,I4) = aa_nat_complex(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_9201_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(D),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_real(C,I4) = aa_nat_real(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_9202_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B4: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_9203_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B4: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_9204_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B4: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_9205_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_rl(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rm(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_9206_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_rn(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rp(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_9207_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( sup_sup_set_rat(set_ord_lessThan_rat(L),set_or633870826150836451st_rat(L,U)) = set_ord_atMost_rat(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_9208_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),U)
     => ( sup_sup_set_num(set_ord_lessThan_num(L),set_or7049704709247886629st_num(L,U)) = set_ord_atMost_num(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_9209_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_set_nat(set_ord_atMost_nat,U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_9210_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( sup_sup_set_int(set_ord_lessThan_int(L),set_or1266510415728281911st_int(L,U)) = set_ord_atMost_int(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_9211_ivl__disj__un__one_I4_J,axiom,
    ! [L: real,U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),U)
     => ( sup_sup_set_real(set_or5984915006950818249n_real(L),set_or1222579329274155063t_real(L,U)) = set_ord_atMost_real(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_9212_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: $o] : sup_sup_set_o(set_ord_lessThan_o((U)),insert_o((U),bot_bot_set_o)) = set_ord_atMost_o((U)) ).

% ivl_disj_un_singleton(2)
tff(fact_9213_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] : sup_sup_set_int(set_ord_lessThan_int(U),insert_int(U,bot_bot_set_int)) = set_ord_atMost_int(U) ).

% ivl_disj_un_singleton(2)
tff(fact_9214_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)),insert_nat(U,bot_bot_set_nat)) = aa_nat_set_nat(set_ord_atMost_nat,U) ).

% ivl_disj_un_singleton(2)
tff(fact_9215_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: real] : sup_sup_set_real(set_or5984915006950818249n_real(U),insert_real(U,bot_bot_set_real)) = set_ord_atMost_real(U) ).

% ivl_disj_un_singleton(2)
tff(fact_9216_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,I3)),aa_real_real(G,I3)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9217_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,I3)),aa_complex_real(G,I3)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9218_prod__mono__strict,axiom,
    ! [A4: set_o,F: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I3)))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(I3))),aa_o_real(G,(I3))) ) )
       => ( ( A4 != bot_bot_set_o )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups234877984723959775o_real(F,A4)),groups234877984723959775o_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9219_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,I3)),aa_nat_real(G,I3)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups129246275422532515t_real(F,A4)),groups129246275422532515t_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9220_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,I3)),aa_int_real(G,I3)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9221_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9222_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9223_prod__mono__strict,axiom,
    ! [A4: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3)))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) ) )
       => ( ( A4 != bot_bot_set_o )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2869687844427037835_o_rat(F,A4)),groups2869687844427037835_o_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9224_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9225_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_9226_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups3455450783089532116nteger(F),A4))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9227_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A4))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9228_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups1230400874837758585nteger(F,A4))
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Pro4764841431850000930nteger(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9229_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A4))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9230_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups4077766827762148844at_nat(F,A4))
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_Pro4243173770044397973at_nat(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9231_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A4))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A4)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9232_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups4075276357253098568at_int(F,A4))
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Pro4240683299535347697at_int(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9233_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( 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(groups708209901874060359at_nat(F),A4))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9234_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A4))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9235_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups1705073143266064639nt_int(F,A4))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_9236_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rq,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_9237_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_real,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups766887009212190081x_real(G,insert_complex(X3,A4)) = aa_real_real(times_times_real(aa_complex_real(G,X3)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_9238_prod_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_real,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups234877984723959775o_real(G,insert_o((X3),A4)) = aa_real_real(times_times_real(aa_o_real(G,(X3))),groups234877984723959775o_real(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_9239_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_real,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups2316167850115554303t_real(G,insert_int(X3,A4)) = aa_real_real(times_times_real(aa_int_real(G,X3)),groups2316167850115554303t_real(G,minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_9240_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G: fun_nat_real,X3: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups129246275422532515t_real(G,insert_nat(X3,A4)) = aa_real_real(times_times_real(aa_nat_real(G,X3)),groups129246275422532515t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_9241_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_rat,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups225925009352817453ex_rat(G,insert_complex(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G,X3)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_9242_prod_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_rat,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups2869687844427037835_o_rat(G,insert_o((X3),A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X3))),groups2869687844427037835_o_rat(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_9243_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_rat,X3: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(G,insert_int(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X3)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_9244_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G: fun_nat_rat,X3: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups73079841787564623at_rat(G,insert_nat(X3,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X3)),groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_9245_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_nat,X3: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups861055069439313189ex_nat(G,insert_complex(X3,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,X3)),groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_9246_prod_Oinsert__remove,axiom,
    ! [A4: set_o,G: fun_o_nat,X3: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups3504817904513533571_o_nat(G,insert_o((X3),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(X3))),groups3504817904513533571_o_nat(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_9247_prod_Oremove,axiom,
    ! [A4: set_real,X3: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => ( groups1681761925125756287l_real(G,A4) = aa_real_real(times_times_real(aa_real_real(G,X3)),groups1681761925125756287l_real(G,minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_9248_prod_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => ( groups766887009212190081x_real(G,A4) = aa_real_real(times_times_real(aa_complex_real(G,X3)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_9249_prod_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X3)),A4)
       => ( groups234877984723959775o_real(G,A4) = aa_real_real(times_times_real(aa_o_real(G,(X3))),groups234877984723959775o_real(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_9250_prod_Oremove,axiom,
    ! [A4: set_int,X3: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(X3),A4)
       => ( groups2316167850115554303t_real(G,A4) = aa_real_real(times_times_real(aa_int_real(G,X3)),groups2316167850115554303t_real(G,minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_9251_prod_Oremove,axiom,
    ! [A4: set_nat,X3: nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(X3),A4)
       => ( groups129246275422532515t_real(G,A4) = aa_real_real(times_times_real(aa_nat_real(G,X3)),groups129246275422532515t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_9252_prod_Oremove,axiom,
    ! [A4: set_real,X3: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(X3),A4)
       => ( groups4061424788464935467al_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G,X3)),groups4061424788464935467al_rat(G,minus_minus_set_real(A4,insert_real(X3,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_9253_prod_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(X3),A4)
       => ( groups225925009352817453ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G,X3)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X3,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_9254_prod_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X3)),A4)
       => ( groups2869687844427037835_o_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X3))),groups2869687844427037835_o_rat(G,minus_minus_set_o(A4,insert_o((X3),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_9255_prod_Oremove,axiom,
    ! [A4: set_int,X3: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(X3),A4)
       => ( groups1072433553688619179nt_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X3)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A4,insert_int(X3,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_9256_prod_Oremove,axiom,
    ! [A4: set_nat,X3: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(X3),A4)
       => ( groups73079841787564623at_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X3)),groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X3,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_9257_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups766887009212190081x_real(G,sup_sup_set_complex(A4,B4)) = aa_real_real(times_times_real(groups766887009212190081x_real(G,A4)),groups766887009212190081x_real(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9258_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups234877984723959775o_real(G,sup_sup_set_o(A4,B4)) = aa_real_real(times_times_real(groups234877984723959775o_real(G,A4)),groups234877984723959775o_real(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9259_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( groups129246275422532515t_real(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_real_real(times_times_real(groups129246275422532515t_real(G,A4)),groups129246275422532515t_real(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9260_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( inf_inf_set_int(A4,B4) = bot_bot_set_int )
         => ( groups2316167850115554303t_real(G,sup_sup_set_int(A4,B4)) = aa_real_real(times_times_real(groups2316167850115554303t_real(G,A4)),groups2316167850115554303t_real(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9261_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups225925009352817453ex_rat(G,sup_sup_set_complex(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G,A4)),groups225925009352817453ex_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9262_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups2869687844427037835_o_rat(G,sup_sup_set_o(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2869687844427037835_o_rat(G,A4)),groups2869687844427037835_o_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9263_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,A4)),groups73079841787564623at_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9264_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( inf_inf_set_int(A4,B4) = bot_bot_set_int )
         => ( groups1072433553688619179nt_rat(G,sup_sup_set_int(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A4)),groups1072433553688619179nt_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9265_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ( inf_inf_set_complex(A4,B4) = bot_bot_set_complex )
         => ( groups861055069439313189ex_nat(G,sup_sup_set_complex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,A4)),groups861055069439313189ex_nat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9266_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups3504817904513533571_o_nat(G,sup_sup_set_o(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3504817904513533571_o_nat(G,A4)),groups3504817904513533571_o_nat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9267_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rr,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_9268_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rr,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9269_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(times_times_real(groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_9270_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_9271_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_9272_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_9273_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_9274_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X3: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X3,Xa2),Xb),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa2),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X3,Xa2),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_9275_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_kz(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(times_times_real(aa_real_real(B,A)),groups1681761925125756287l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1681761925125756287l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% prod.delta_remove
tff(fact_9276_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_la(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_real_real(times_times_real(aa_complex_real(B,A)),groups766887009212190081x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups766887009212190081x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_9277_prod_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: fun_o_real,C: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( groups234877984723959775o_real(aa_fun688608109223240316o_real(aa_fun1133276423545935902o_real(aTP_Lamm_lh((A)),B),C),S2) = $ite(aa_set_o_o(member_o((A)),S2),aa_real_real(times_times_real(aa_o_real(B,(A))),groups234877984723959775o_real(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))),groups234877984723959775o_real(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_9278_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_real,C: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_rs(A),B),C),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_real_real(times_times_real(aa_int_real(B,A)),groups2316167850115554303t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))),groups2316167850115554303t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_9279_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real,C: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rt(A),B),C),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_real_real(times_times_real(aa_nat_real(B,A)),groups129246275422532515t_real(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S2),insert_nat(A,bot_bot_set_nat)))),groups129246275422532515t_real(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S2),insert_nat(A,bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_9280_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lb(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(B,A)),groups4061424788464935467al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups4061424788464935467al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% prod.delta_remove
tff(fact_9281_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_lc(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(B,A)),groups225925009352817453ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups225925009352817453ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_9282_prod_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: fun_o_rat,C: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_li((A)),B),C),S2) = $ite(aa_set_o_o(member_o((A)),S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(B,(A))),groups2869687844427037835_o_rat(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))),groups2869687844427037835_o_rat(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_9283_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ru(A),B),C),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(B,A)),groups1072433553688619179nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))),groups1072433553688619179nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_9284_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat,C: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rv(A),B),C),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(B,A)),groups73079841787564623at_rat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S2),insert_nat(A,bot_bot_set_nat)))),groups73079841787564623at_rat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S2),insert_nat(A,bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_9285_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,K2: nat] :
      ( ! [W2: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),W2),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_complex(C,K2) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_9286_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,K2: nat] :
      ( ! [W2: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),W2),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_real(C,K2) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_9287_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( ! [X4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_complex(C,I4) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_9288_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( ! [X4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_real(C,I4) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_9289_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9290_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9291_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_is(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9292_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_il(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9293_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9294_sum__up__index__split,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9295_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9296_sum__up__index__split,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9297_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_rw(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9298_gbinomial__parallel__sum,axiom,
    ! [A: rat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_rx(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9299_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_ry(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9300_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_rz(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_sb(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_9301_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_rz(Nb)))) = groups6591440286371151544t_real(aTP_Lamm_sd(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_9302_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_9303_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B3)) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9304_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_o_real(F,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(A3))) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups234877984723959775o_real(F,A4)),groups234877984723959775o_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9305_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9306_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9307_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3)) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9308_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(A3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A4)),groups2869687844427037835_o_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9309_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9310_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9311_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,B3)) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9312_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(A3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A4)),groups3502327434004483295_o_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9313_prod__diff1,axiom,
    ! [A4: set_real,F: fun_real_complex,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( groups713298508707869441omplex(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A4)),aa_real_complex(F,A)),groups713298508707869441omplex(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9314_prod__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A4)),aa_complex_complex(F,A)),groups3708469109370488835omplex(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9315_prod__diff1,axiom,
    ! [A4: set_o,F: fun_o_complex,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( aa_o_complex(F,(A)) != zero_zero_complex )
       => ( groups4859619685533338977omplex(F,minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),aa_complex_complex(divide1717551699836669952omplex(groups4859619685533338977omplex(F,A4)),aa_o_complex(F,(A))),groups4859619685533338977omplex(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9316_prod__diff1,axiom,
    ! [A4: set_int,F: fun_int_complex,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( groups7440179247065528705omplex(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A4)),aa_int_complex(F,A)),groups7440179247065528705omplex(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9317_prod__diff1,axiom,
    ! [A4: set_nat,F: fun_nat_complex,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( groups6464643781859351333omplex(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A4),aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(F,A4)),aa_nat_complex(F,A)),groups6464643781859351333omplex(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9318_prod__diff1,axiom,
    ! [A4: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( aa_real_real(F,A) != zero_zero_real )
       => ( groups1681761925125756287l_real(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_real_real(divide_divide_real(groups1681761925125756287l_real(F,A4)),aa_real_real(F,A)),groups1681761925125756287l_real(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9319_prod__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( aa_complex_real(F,A) != zero_zero_real )
       => ( groups766887009212190081x_real(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_real_real(divide_divide_real(groups766887009212190081x_real(F,A4)),aa_complex_real(F,A)),groups766887009212190081x_real(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9320_prod__diff1,axiom,
    ! [A4: set_o,F: fun_o_real,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( aa_o_real(F,(A)) != zero_zero_real )
       => ( groups234877984723959775o_real(F,minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),aa_real_real(divide_divide_real(groups234877984723959775o_real(F,A4)),aa_o_real(F,(A))),groups234877984723959775o_real(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9321_prod__diff1,axiom,
    ! [A4: set_int,F: fun_int_real,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( aa_int_real(F,A) != zero_zero_real )
       => ( groups2316167850115554303t_real(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),aa_real_real(divide_divide_real(groups2316167850115554303t_real(F,A4)),aa_int_real(F,A)),groups2316167850115554303t_real(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9322_prod__diff1,axiom,
    ! [A4: set_nat,F: fun_nat_real,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_nat_real(F,A) != zero_zero_real )
       => ( groups129246275422532515t_real(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A4),aa_real_real(divide_divide_real(groups129246275422532515t_real(F,A4)),aa_nat_real(F,A)),groups129246275422532515t_real(F,A4)) ) ) ) ).

% prod_diff1
tff(fact_9323_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_se(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_9324_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_sf(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_9325_sum__gp__basic,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X3)),groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X3),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9326_sum__gp__basic,axiom,
    ! [X3: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3)),groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X3),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9327_sum__gp__basic,axiom,
    ! [X3: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X3)),groups3539618377306564664at_int(power_power_int(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X3),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9328_sum__gp__basic,axiom,
    ! [X3: real,Nb: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X3)),groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9329_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K2: nat,Nb: nat] :
      ( ( aa_nat_complex(C,K2) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_sg(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_9330_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K2: nat,Nb: nat] :
      ( ( aa_nat_real(C,K2) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_sh(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_9331_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_sg(C),Nb)))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
          & ( aa_nat_complex(C,I4) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_9332_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_sh(C),Nb)))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
          & ( aa_nat_real(C,I4) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_9333_pochhammer__Suc__prod,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_si(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_9334_pochhammer__Suc__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aTP_Lamm_sj(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_9335_pochhammer__Suc__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sk,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_9336_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_sl(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_9337_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,Nb: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9338_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,Nb: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9339_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,Nb: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9340_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,Nb: nat] :
      ( ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z4),A)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9341_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,Nb: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9342_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,Nb: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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,Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9343_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,Nb: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9344_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,Nb: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z4),A)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9345_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,X3: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X3),Mb)),groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9346_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,X3: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),Mb)),groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9347_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,X3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(power_power_int(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X3),Mb)),groups3539618377306564664at_int(power_power_int(X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9348_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(X3),Mb)),groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9349_pochhammer__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_so(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_9350_pochhammer__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_sp(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_9351_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sq(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_9352_pochhammer__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sr(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_9353_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_9354_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ss(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_sb(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_9355_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ss(Nb)))) = groups6591440286371151544t_real(aTP_Lamm_sd(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_9356_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_9357_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_st(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9358_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Nb)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9359_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_qu(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_9360_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_qv(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_9361_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qw(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_9362_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_qx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_9363_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_su(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_9364_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_sv(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_9365_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_sw(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_9366_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_sx(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_9367_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_mc(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9368_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_md(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9369_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_me(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9370_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_mf(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9371_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,X3: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_complex(A,I3) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(B),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_sz(A),B),X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9372_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_rat,Nb: nat,B: fun_nat_rat,X3: rat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_rat(A,I3) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(B),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_tb(A),B),X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9373_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,X3: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_int(A,I3) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(B),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_td(A),B),X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9374_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,X3: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_real(A,I3) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(times_times_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(B),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_tf(A),B),X3),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9375_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_so(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_9376_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_sp(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_9377_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sq(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_9378_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sr(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_9379_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Nb: nat,K2: complex] :
      ( ! [X4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(C),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K2
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K2 )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))
           => ( aa_nat_complex(C,X4) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_9380_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Nb: nat,K2: real] :
      ( ! [X4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(C),X4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K2
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K2 )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))
           => ( aa_nat_real(C,X4) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_9381_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_tg(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Mb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_9382_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_th(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Mb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_9383_gbinomial__sum__lower__neg,axiom,
    ! [A: real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_ti(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Mb)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_9384_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tj(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9385_binomial__ring,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tk(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9386_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tl(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9387_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_st(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9388_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(plus_plus_real(A),B)),Nb) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tm(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9389_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tn(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_9390_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_to(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_9391_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(A),B)),Nb) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tp(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_9392_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,X3: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tq(A),X3)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tq(B),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ts(A),B),X3)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9393_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tt,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_9394_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(aTP_Lamm_tu(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_9395_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,groups73079841787564623at_rat(aTP_Lamm_tv(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_9396_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K2)) = aa_real_real(divide_divide_real(groups129246275422532515t_real(aTP_Lamm_tw(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_9397_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tx,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_9398_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,groups705719431365010083at_int(aTP_Lamm_ty(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_9399_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rf(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9400_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ua(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rj(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9401_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ub(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rh(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9402_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uc(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rd(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9403_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: complex,X3: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ud(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ue(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_9404_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: rat,X3: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uf(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ug(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_9405_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: real,X3: real,Y: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_uh(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ui(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_9406_root__polyfun,axiom,
    ! [Nb: nat,Z: int,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_int(power_power_int(Z),Nb) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_uj(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_9407_root__polyfun,axiom,
    ! [Nb: nat,Z: complex,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_complex(power_power_complex(Z),Nb) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_uk(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_9408_root__polyfun,axiom,
    ! [Nb: nat,Z: code_integer,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),Nb) = A )
      <=> ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_ul(Nb),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_9409_root__polyfun,axiom,
    ! [Nb: nat,Z: rat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_rat(power_power_rat(Z),Nb) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_um(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_9410_root__polyfun,axiom,
    ! [Nb: nat,Z: real,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_real(power_power_real(Z),Nb) = A )
      <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_un(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_9411_sum__gp0,axiom,
    ! [X3: complex,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(X3 = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X3),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),X3))) ).

% sum_gp0
tff(fact_9412_sum__gp0,axiom,
    ! [X3: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(X3 = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X3),aa_nat_nat(suc,Nb)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X3))) ).

% sum_gp0
tff(fact_9413_sum__gp0,axiom,
    ! [X3: real,Nb: nat] :
      groups6591440286371151544t_real(power_power_real(X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(X3 = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),X3))) ).

% sum_gp0
tff(fact_9414_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_uo(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_9415_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_up(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_9416_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_uq(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_9417_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ur,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_9418_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_us,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_9419_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_ut(Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_9420_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_uu(Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_9421_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uv,Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_9422_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: complex,X3: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ud(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_uw(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_9423_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: rat,X3: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uf(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ux(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_9424_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: real,X3: real,Y: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_uh(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_uy(Mb),A),X3),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_9425_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_complex,X3: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X3),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_va(Nb),A),X3),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_9426_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_rat,X3: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_vc(Nb),A),X3),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_9427_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_int,X3: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sn(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_ve(Nb),A),X3),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_9428_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_real,X3: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(minus_minus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(A),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(X3),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_vg(Nb),A),X3),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_9429_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% binomial_r_part_sum
tff(fact_9430_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vh,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_9431_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_vi(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_9432_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_vj,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_9433_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vk,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_9434_Maclaurin__sin__bound,axiom,
    ! [X3: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,X3)),groups6591440286371151544t_real(aTP_Lamm_pe(X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,X3)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_9435_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9436_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D4: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E2)
         => ( aa_real_o(P,D4)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_9437_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_om,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_9438_exp__plus__inverse__exp,axiom,
    ! [X3: 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(plus_plus_real(aa_real_real(exp_real,X3)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X3)))) ).

% exp_plus_inverse_exp
tff(fact_9439_plus__inverse__ge__2,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(plus_plus_real(X3),aa_real_real(inverse_inverse_real,X3))) ) ).

% plus_inverse_ge_2
tff(fact_9440_real__inv__sqrt__pow2,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X3) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9441_tan__cot,axiom,
    ! [X3: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X3)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X3)) ).

% tan_cot
tff(fact_9442_real__le__x__sinh,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X3)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X3)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9443_real__le__abs__sinh,axiom,
    ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X3)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X3)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9444_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T4)),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9445_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_vl,Y)) ).

% arctan_def
tff(fact_9446_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_vm,Y)) ).

% arcsin_def
tff(fact_9447_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9448_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9449_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(divide_divide_real(A),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9450_complex__norm,axiom,
    ! [X3: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(X3),Y)) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9451_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_vn) ).

% pi_half
tff(fact_9452_pi__def,axiom,
    pi = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_vn)) ).

% pi_def
tff(fact_9453_cosh__ln__real,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X3)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X3),aa_real_real(inverse_inverse_real,X3))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9454_sinh__ln__real,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X3)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X3),aa_real_real(inverse_inverse_real,X3))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9455_powr__int,axiom,
    ! [X3: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(powr_real(X3),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(power_power_real(X3),aa_int_nat(nat2,I)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(X3),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_9456_divide__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_9457_arctan__inverse,axiom,
    ! [X3: real] :
      ( ( X3 != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),X3)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,X3)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X3)) ) ) ).

% arctan_inverse
tff(fact_9458_nat__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_numeral
tff(fact_9459_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9460_nat__neg__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_9461_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V2)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_9462_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9463_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X3: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9464_nat__ceiling__le__eq,axiom,
    ! [X3: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X3))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_9465_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_9466_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V2)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V2)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_9467_numeral__power__less__nat__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3)),Nb)),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,X3)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9468_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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,X3)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9469_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,Nb: 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,X3)),Nb))
    <=> 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,X3)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9470_numeral__power__le__nat__cancel__iff,axiom,
    ! [X3: num,Nb: 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,X3)),Nb)),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,X3)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9471_nat__numeral__as__int,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,X) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X)) ).

% nat_numeral_as_int
tff(fact_9472_nat__mono,axiom,
    ! [X3: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X3)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_9473_nat__le__iff,axiom,
    ! [X3: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X3)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_9474_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_9475_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_9476_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9477_le__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K2) ) ) ).

% le_nat_iff
tff(fact_9478_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_9479_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_9480_nat__abs__triangle__ineq,axiom,
    ! [K2: 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,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_9481_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_9482_floor__eq3,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X3)) = Nb ) ) ) ).

% floor_eq3
tff(fact_9483_le__nat__floor,axiom,
    ! [X3: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X3)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9484_divide__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)))))) ) ).

% divide_int_def
tff(fact_9485_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_9486_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,X3: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb)) = X3 )
     => ( ( X3 = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9487_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_9488_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_9489_floor__eq4,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X3)) = Nb ) ) ) ).

% floor_eq4
tff(fact_9490_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(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))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9491_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9492_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q2: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
         => ~ ! [R3: int,Q2: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9493_even__nat__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( 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,K2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ) ).

% even_nat_iff
tff(fact_9494_powr__real__of__int,axiom,
    ! [X3: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => ( aa_real_real(powr_real(X3),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(power_power_real(X3),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(X3),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9495_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9496_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9497_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9498_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9499_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9500_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_9501_xor__nat__numerals_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3)) ).

% xor_nat_numerals(4)
tff(fact_9502_xor__nat__numerals_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% xor_nat_numerals(3)
tff(fact_9503_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_9504_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9505_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9506_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9507_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9508_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9509_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != numera6690914467698888265omplex(W) ).

% complex_i_not_numeral
tff(fact_9510_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_9511_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_9512_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9513_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9514_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9515_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9516_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_vo(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aTP_Lamm_ev(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9517_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9518_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9519_XOR__upper,axiom,
    ! [X3: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X3),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9520_xor__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9521_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_ev(Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vp(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9522_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = $ite(
        K2 = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K2),
          $ite(
            K2 = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9523_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9524_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_9525_real__root__Suc__0,axiom,
    ! [X3: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X3) = X3 ).

% real_root_Suc_0
tff(fact_9526_real__root__pow__pos2,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),Nb) = X3 ) ) ) ).

% real_root_pow_pos2
tff(fact_9527_or__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9528_and__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9529_real__root__power,axiom,
    ! [Nb: nat,X3: real,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(X3),K2)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),K2) ) ) ).

% real_root_power
tff(fact_9530_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9531_not__int__div__2,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9532_even__not__iff__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_not_iff_int
tff(fact_9533_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9534_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_9535_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9536_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9537_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9538_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9539_int__numeral__not__or__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_9540_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9541_real__root__decreasing,axiom,
    ! [Nb: nat,N4: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X3)),aa_real_real(aa_nat_fun_real_real(root,Nb),X3)) ) ) ) ).

% real_root_decreasing
tff(fact_9542_real__root__pow__pos,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),Nb) = X3 ) ) ) ).

% real_root_pow_pos
tff(fact_9543_real__root__power__cancel,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(X3),Nb)) = X3 ) ) ) ).

% real_root_power_cancel
tff(fact_9544_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(power_power_real(Y),Nb) = X3 )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),X3) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9545_odd__real__root__pow,axiom,
    ! [Nb: nat,X3: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),Nb) = X3 ) ) ).

% odd_real_root_pow
tff(fact_9546_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,X3: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(power_power_real(Y),Nb) = X3 )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),X3) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9547_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,X3: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(X3),Nb)) = X3 ) ) ).

% odd_real_root_power_cancel
tff(fact_9548_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9549_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_9550_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9551_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9552_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9553_real__root__increasing,axiom,
    ! [Nb: nat,N4: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_real_real(aa_nat_fun_real_real(root,N4),X3)) ) ) ) ) ).

% real_root_increasing
tff(fact_9554_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9555_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9556_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9557_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_9558_sgn__power__root,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X3))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X3))),Nb)) = X3 ) ) ).

% sgn_power_root
tff(fact_9559_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9560_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,X3: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),X3))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y4: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y4)),Nb)) = X3 )
             => aa_real_o(P,Y4) ) ) ) ) ).

% split_root
tff(fact_9561_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9562_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9563_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9564_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9565_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9566_not__int__rec,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = 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))),K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9567_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9568_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9569_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9570_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9571_atLeastLessThan0,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9572_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_9573_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Mb),K2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9574_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q3)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9575_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9576_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9577_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9578_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9579_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9580_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9581_push__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9582_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K2: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(numeral_numeral_nat,K2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K2)),insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(pred_numeral,K2))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9583_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_minus_one
tff(fact_9584_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Nb)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9585_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K2)) = 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))),K2)),one_one_nat) ).

% sum_power2
tff(fact_9586_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I3)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vq(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9587_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M4)
         => ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X6,M4)),aa_nat_real(X6,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9588_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_9589_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9590_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9591_complex__Re__numeral,axiom,
    ! [V2: num] : aa_complex_real(re,numera6690914467698888265omplex(V2)) = aa_num_real(numeral_numeral_real,V2) ).

% complex_Re_numeral
tff(fact_9592_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(re,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Re_divide_numeral
tff(fact_9593_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9594_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_9595_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(times_times_real($ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9596_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(times_times_real($ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9597_Complex__divide,axiom,
    ! [X3: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X3),Y) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,X3)),aa_complex_real(re,Y))),aa_real_real(times_times_real(aa_complex_real(im,X3)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,X3)),aa_complex_real(re,Y))),aa_real_real(times_times_real(aa_complex_real(re,X3)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9598_Im__power__real,axiom,
    ! [X3: complex,Nb: nat] :
      ( ( aa_complex_real(im,X3) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(X3),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_9599_complex__Im__numeral,axiom,
    ! [V2: num] : aa_complex_real(im,numera6690914467698888265omplex(V2)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9600_Re__power__real,axiom,
    ! [X3: complex,Nb: nat] :
      ( ( aa_complex_real(im,X3) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(X3),Nb)) = aa_nat_real(power_power_real(aa_complex_real(re,X3)),Nb) ) ) ).

% Re_power_real
tff(fact_9601_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(im,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Im_divide_numeral
tff(fact_9602_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9603_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9604_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9605_Im__power2,axiom,
    ! [X3: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,X3))),aa_complex_real(im,X3)) ).

% Im_power2
tff(fact_9606_Re__power2,axiom,
    ! [X3: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_complex_real(re,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9607_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9608_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9609_inverse__complex_Osimps_I1_J,axiom,
    ! [X3: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,X3)) = aa_real_real(divide_divide_real(aa_complex_real(re,X3)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9610_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(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9611_Re__divide,axiom,
    ! [X3: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(X3),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,X3)),aa_complex_real(re,Y))),aa_real_real(times_times_real(aa_complex_real(im,X3)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9612_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9613_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9614_inverse__complex_Osimps_I2_J,axiom,
    ! [X3: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,X3)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,X3))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9615_Im__divide,axiom,
    ! [X3: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(X3),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,X3)),aa_complex_real(re,Y))),aa_real_real(times_times_real(aa_complex_real(re,X3)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9616_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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(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_9617_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9618_inverse__complex_Ocode,axiom,
    ! [X3: complex] : aa_complex_complex(invers8013647133539491842omplex,X3) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_complex_real(re,X3)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,X3))),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9619_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9620_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9621_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9622_complex__cnj__power,axiom,
    ! [X3: complex,Nb: nat] : aa_complex_complex(cnj,aa_nat_complex(power_power_complex(X3),Nb)) = aa_nat_complex(power_power_complex(aa_complex_complex(cnj,X3)),Nb) ).

% complex_cnj_power
tff(fact_9623_complex__cnj__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,numera6690914467698888265omplex(W)) = numera6690914467698888265omplex(W) ).

% complex_cnj_numeral
tff(fact_9624_complex__cnj__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_cnj_neg_numeral
tff(fact_9625_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9626_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9627_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9628_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9629_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9630_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9631_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_fy(L),Qr) ).

% divmod_step_integer_def
tff(fact_9632_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9633_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eq,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9634_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9635_divmod__integer_H__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_integer'_def
tff(fact_9636_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_vr),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9637_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_ch),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9638_card__less__Suc2,axiom,
    ! [M6: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_vs(M6),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_vt(M6),I))) ) ) ).

% card_less_Suc2
tff(fact_9639_card__less__Suc,axiom,
    ! [M6: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_vs(M6),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_vt(M6),I))) ) ) ).

% card_less_Suc
tff(fact_9640_card__less,axiom,
    ! [M6: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_vt(M6),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9641_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_nat_set_nat(set_or4665077453230672383an_nat(K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))))
     => ( A4 = aa_nat_set_nat(set_or4665077453230672383an_nat(K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9642_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9643_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_vu,Nb)),Mb) ).

% max_Suc1
tff(fact_9644_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_vv,Nb)),Mb) ).

% max_Suc2
tff(fact_9645_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9646_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mg),S2)) ).

% card_sum_le_nat_sum
tff(fact_9647_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vp(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9648_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aTP_Lamm_ev(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9649_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_mg),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_9650_integer__of__int__code,axiom,
    ! [K2: int] :
      aa_int_Code_integer(code_integer_of_int,K2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K2))),
        $ite(
          K2 = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),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_9651_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_9652_bit__cut__integer__def,axiom,
    ! [K2: code_integer] : code_bit_cut_integer(K2) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K2)) ).

% bit_cut_integer_def
tff(fact_9653_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_num_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_vw,code_divmod_integer(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9654_divmod__integer__def,axiom,
    ! [K2: code_integer,L: code_integer] : code_divmod_integer(K2,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K2,L),modulo364778990260209775nteger(K2,L)) ).

% divmod_integer_def
tff(fact_9655_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      code_bit_cut_integer(K2) = $ite(K2 = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_vx(K2),code_divmod_abs(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9656_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Code_integer_nat(code_nat_of_integer,K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_vy,code_divmod_integer(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9657_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_int_of_integer(K2) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2))),
        $ite(K2 = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_vz,code_divmod_integer(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9658_int__of__integer__numeral,axiom,
    ! [K2: num] : code_int_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K2)) = aa_num_int(numeral_numeral_int,K2) ).

% int_of_integer_numeral
tff(fact_9659_nat__of__integer__code__post_I3_J,axiom,
    ! [K2: num] : aa_Code_integer_nat(code_nat_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_of_integer_code_post(3)
tff(fact_9660_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,abs_abs_Code_integer(J)) ).

% divmod_abs_code(5)
tff(fact_9661_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_9662_divmod__abs__def,axiom,
    ! [K2: code_integer,L: code_integer] : code_divmod_abs(K2,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_9663_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_mg),Nat) ).

% pred_def
tff(fact_9664_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = 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),K2),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_wa(L),code_divmod_abs(K2,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_wb(L),code_divmod_abs(K2,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9665_bezw__0,axiom,
    ! [X3: nat] : bezw(X3,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_9666_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9667_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9668_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9669_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9670_drop__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9671_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9672_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9673_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9674_Suc__0__mod__numeral,axiom,
    ! [K2: 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,K2)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_mod_numeral
tff(fact_9675_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,Mb: nat] :
      aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),Mb)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_9676_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X3),Xa2) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa2)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X3)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9677_Suc__0__div__numeral,axiom,
    ! [K2: 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,K2)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_div_numeral
tff(fact_9678_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5) ) ).

% Inf_nat_def1
tff(fact_9679_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_9680_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_9681_minus__numeral__mod__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9682_numeral__mod__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(Mb,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9683_bezw__non__0,axiom,
    ! [Y: nat,X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(X3,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Y))))) ) ) ).

% bezw_non_0
tff(fact_9684_bezw_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X3,Xa2) = Y )
     => ( Y = $ite(Xa2 = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa2)))))) ) ) ).

% bezw.elims
tff(fact_9685_bezw_Osimps,axiom,
    ! [X3: nat,Y: nat] :
      bezw(X3,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Y)))))) ).

% bezw.simps
tff(fact_9686_bezw_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X3,Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa2)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_9687_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X3),Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa2)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X3)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9688_vebt__maxt_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X3) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X3)
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      some_nat(one_one_nat),
                      $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9689_vebt__mint_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X3) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X3)
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      (A3),
                      some_nat(zero_zero_nat),
                      $ite((B3),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9690_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
         => aa_nat_o(P,K) )
     => ( ! [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
           => ( ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),I2)
                 => aa_nat_o(P,I2) )
             => aa_nat_o(P,K) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_9691_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X3)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X3)
       => ( ( ( X3 = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( X3 = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( X3 = vEBT_Leaf((Uu),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9692_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X3)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X3)
       => ( ( ( X3 = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9693_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X3)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X3)
       => ( ! [Uv: $o] :
              ( ( X3 = vEBT_Leaf($true,(Uv)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu: $o] :
                ( ( X3 = vEBT_Leaf((Uu),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9694_xor__minus__numerals_I2_J,axiom,
    ! [K2: int,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),aa_num_int(neg_numeral_sub_int(Nb),one))) ).

% xor_minus_numerals(2)
tff(fact_9695_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K2: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K2) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(neg_numeral_sub_int(Nb),one)),K2)) ).

% xor_minus_numerals(1)
tff(fact_9696_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9697_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9698_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_eq,aTP_Lamm_mb) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9699_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_wc(L),code_divmod_abs(K2,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9700_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X3)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_we),Xa2),X3)) ).

% times_int.abs_eq
tff(fact_9701_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_9702_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X5: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3)) ).

% eq_Abs_Integ
tff(fact_9703_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9704_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).

% int_def
tff(fact_9705_uminus__int_Oabs__eq,axiom,
    ! [X3: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X3)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_wf),X3)) ).

% uminus_int.abs_eq
tff(fact_9706_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9707_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X3))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_wh),Xa2),X3) ) ).

% less_int.abs_eq
tff(fact_9708_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X3))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_wj),Xa2),X3) ) ).

% less_eq_int.abs_eq
tff(fact_9709_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X3)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_wl),Xa2),X3)) ).

% plus_int.abs_eq
tff(fact_9710_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X3)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_wn),Xa2),X3)) ).

% minus_int.abs_eq
tff(fact_9711_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9712_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_9713_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wo)) ).

% pred_nat_def
tff(fact_9714_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9715_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q3)) = Q3 ).

% num_of_nat_numeral_eq
tff(fact_9716_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9717_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9718_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9719_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9720_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9721_less__eq__int_Orep__eq,axiom,
    ! [X3: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa2)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_wj),aa_int1417691210696471063at_nat(rep_Integ,X3)),aa_int1417691210696471063at_nat(rep_Integ,Xa2)) ) ).

% less_eq_int.rep_eq
tff(fact_9722_less__int_Orep__eq,axiom,
    ! [X3: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa2)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_wh),aa_int1417691210696471063at_nat(rep_Integ,X3)),aa_int1417691210696471063at_nat(rep_Integ,Xa2)) ) ).

% less_int.rep_eq
tff(fact_9723_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_wf)) ).

% uminus_int_def
tff(fact_9724_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_we)) ).

% times_int_def
tff(fact_9725_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_wn)) ).

% minus_int_def
tff(fact_9726_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_wl)) ).

% plus_int_def
tff(fact_9727_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(gcd_Gcd_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M6),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9728_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9729_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_wp(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9730_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X3: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wq,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X3),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9731_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_9732_bij__betw__Suc,axiom,
    ! [M6: set_nat,N4: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N4)
    <=> ( image_nat_nat(suc,M6) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9733_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9734_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9735_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9736_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9737_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A4)) ).

% zero_notin_Suc_image
tff(fact_9738_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_wr,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9739_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_9740_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb) ).

% image_Suc_lessThan
tff(fact_9741_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9742_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9743_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9744_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9745_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9746_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9747_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9748_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_9749_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_9750_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_9751_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_9752_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_9753_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_9754_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ws,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_9755_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_9756_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ws,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_9757_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9758_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_wt,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_9759_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_9760_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9761_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_9762_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_wu(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9763_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_9764_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q3: num] :
      ( ( bit_and_not_num(Mb,Nb) = some_num(Q3) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9765_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_wv(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9766_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9767_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_9768_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_9769_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_wz,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9770_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_9771_and__not__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( bit_and_not_num(X3,Xa2) = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X3 = one )
           => ( ? [N: num] : Xa2 = aa_num_num(bit0,N)
             => ( Y != some_num(one) ) ) )
         => ( ( ( X3 = one )
             => ( ? [N: num] : Xa2 = aa_num_num(bit1,N)
               => ( Y != none_num ) ) )
           => ( ! [M: num] :
                  ( ( X3 = aa_num_num(bit0,M) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
             => ( ! [M: num] :
                    ( ( X3 = aa_num_num(bit0,M) )
                   => ! [N: num] :
                        ( ( Xa2 = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M,N)) ) ) )
               => ( ! [M: num] :
                      ( ( X3 = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xa2 = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M,N)) ) ) )
                 => ( ! [M: num] :
                        ( ( X3 = aa_num_num(bit1,M) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( X3 = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xa2 = aa_num_num(bit0,N) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_wt,bit_and_not_num(M,N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( X3 = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9772_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9773_and__not__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(5)
tff(fact_9774_and__not__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(9)
tff(fact_9775_and__not__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(6)
tff(fact_9776_root__def,axiom,
    ! [Nb: nat,X3: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),X3) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_xa,Nb)),X3)) ).

% root_def
tff(fact_9777_and__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X3,Xa2) = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( X3 = one )
           => ( ? [N: num] : Xa2 = aa_num_num(bit0,N)
             => ( Y != none_num ) ) )
         => ( ( ( X3 = one )
             => ( ? [N: num] : Xa2 = aa_num_num(bit1,N)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M: num] : X3 = aa_num_num(bit0,M)
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M: num] :
                    ( ( X3 = aa_num_num(bit0,M) )
                   => ! [N: num] :
                        ( ( Xa2 = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N)) ) ) )
               => ( ! [M: num] :
                      ( ( X3 = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xa2 = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N)) ) ) )
                 => ( ( ? [M: num] : X3 = aa_num_num(bit1,M)
                     => ( ( Xa2 = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M: num] :
                          ( ( X3 = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xa2 = aa_num_num(bit0,N) )
                             => ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( X3 = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit1,N) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_wt,bit_un7362597486090784418nd_num(M,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9778_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9779_and__num_Osimps_I1_J,axiom,
    bit_un7362597486090784418nd_num(one,one) = some_num(one) ).

% and_num.simps(1)
tff(fact_9780_and__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_un7362597486090784418nd_num(Mb,Nb)) ).

% and_num.simps(5)
tff(fact_9781_and__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,Mb),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9782_and__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit1,Nb)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9783_and__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit0,Nb)) = none_num ).

% and_num.simps(2)
tff(fact_9784_and__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),one) = none_num ).

% and_num.simps(4)
tff(fact_9785_and__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_un7362597486090784418nd_num(Mb,Nb)) ).

% and_num.simps(6)
tff(fact_9786_and__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_un7362597486090784418nd_num(Mb,Nb)) ).

% and_num.simps(8)
tff(fact_9787_and__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_wt,bit_un7362597486090784418nd_num(Mb,Nb)) ).

% and_num.simps(9)
tff(fact_9788_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = 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))))))))) ).

% card_UNIV_char
tff(fact_9789_xor__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( bit_un2480387367778600638or_num(X3,Xa2) = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X3 = one )
           => ! [N: num] :
                ( ( Xa2 = aa_num_num(bit0,N) )
               => ( Y != some_num(aa_num_num(bit1,N)) ) ) )
         => ( ( ( X3 = one )
             => ! [N: num] :
                  ( ( Xa2 = aa_num_num(bit1,N) )
                 => ( Y != some_num(aa_num_num(bit0,N)) ) ) )
           => ( ! [M: num] :
                  ( ( X3 = aa_num_num(bit0,M) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit1,M)) ) ) )
             => ( ! [M: num] :
                    ( ( X3 = aa_num_num(bit0,M) )
                   => ! [N: num] :
                        ( ( Xa2 = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N)) ) ) )
               => ( ! [M: num] :
                      ( ( X3 = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xa2 = aa_num_num(bit1,N) )
                         => ( Y != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N))) ) ) )
                 => ( ! [M: num] :
                        ( ( X3 = aa_num_num(bit1,M) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( X3 = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xa2 = aa_num_num(bit0,N) )
                             => ( Y != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N))) ) ) )
                     => ~ ! [M: num] :
                            ( ( X3 = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9790_xor__num_Osimps_I1_J,axiom,
    bit_un2480387367778600638or_num(one,one) = none_num ).

% xor_num.simps(1)
tff(fact_9791_xor__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_un2480387367778600638or_num(Mb,Nb)) ).

% xor_num.simps(5)
tff(fact_9792_xor__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_un2480387367778600638or_num(Mb,Nb)) ).

% xor_num.simps(9)
tff(fact_9793_xor__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_un2480387367778600638or_num(one,aa_num_num(bit0,Nb)) = some_num(aa_num_num(bit1,Nb)) ).

% xor_num.simps(2)
tff(fact_9794_xor__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_un2480387367778600638or_num(one,aa_num_num(bit1,Nb)) = some_num(aa_num_num(bit0,Nb)) ).

% xor_num.simps(3)
tff(fact_9795_xor__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit1,Mb)) ).

% xor_num.simps(4)
tff(fact_9796_xor__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% xor_num.simps(7)
tff(fact_9797_xor__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(Mb,Nb))) ).

% xor_num.simps(6)
tff(fact_9798_xor__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(Mb,Nb))) ).

% xor_num.simps(8)
tff(fact_9799_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9800_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9801_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_9802_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9803_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B72))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B62)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B52)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B42)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B32)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B22)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B1)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B0))) ).

% integer_of_char_code
tff(fact_9804_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9805_DERIV__even__real__root,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9806_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K2)),cons_nat(K2,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9807_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K2)),cons_nat(aa_nat_nat(suc,K2),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9808_DERIV__const__average,axiom,
    ! [A: real,B: real,V2: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X5: real] : has_fi5821293074295781190e_real(V2,K2,topolo2177554685111907308n_real(X5,top_top_set_real))
       => ( aa_real_real(V2,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(V2,A)),aa_real_real(V2,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9809_DERIV__pow,axiom,
    ! [Nb: nat,X3: real,Sb: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_xb,Nb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(X3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X3,Sb)) ).

% DERIV_pow
tff(fact_9810_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Mb: real,X3: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(X3,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_xc(G),Nb),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(G,X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Mb),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9811_artanh__real__has__field__derivative,axiom,
    ! [X3: real,A4: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X3,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_9812_DERIV__real__sqrt,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X3))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9813_DERIV__arctan,axiom,
    ! [X3: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X3,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9814_arsinh__real__has__field__derivative,axiom,
    ! [X3: real,A4: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X3,A4)) ).

% arsinh_real_has_field_derivative
tff(fact_9815_DERIV__real__sqrt__generic,axiom,
    ! [X3: real,D2: real] :
      ( ( X3 != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
         => ( D2 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X3))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
           => ( D2 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X3)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D2,topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9816_arcosh__real__has__field__derivative,axiom,
    ! [X3: real,A4: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X3)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X3,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9817_DERIV__real__root,axiom,
    ! [Nb: nat,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9818_DERIV__arccos,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9819_DERIV__arcsin,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9820_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X3: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,X5: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X5),topolo2177554685111907308n_real(X5,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,X3))
            & ( aa_real_real(F,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xd(Diff),X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9821_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X3: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M: nat,X5: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X5),topolo2177554685111907308n_real(X5,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,X3))
          & ( aa_real_real(F,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xd(Diff),X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9822_DERIV__odd__real__root,axiom,
    ! [Nb: nat,X3: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( X3 != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9823_Maclaurin,axiom,
    ! [H: real,Nb: 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),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & 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,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),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(plus_plus_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ox(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9824_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & 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,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),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(plus_plus_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ox(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9825_Maclaurin__minus,axiom,
    ! [H: real,Nb: 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),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & 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,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),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(plus_plus_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ox(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9826_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,X3: 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),Nb)
       => ( ( X3 != zero_zero_real )
         => ( ! [M: nat,X5: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X5),topolo2177554685111907308n_real(X5,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,X3))
                & ( aa_real_real(F,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xd(Diff),X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9827_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,X3: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & 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,X3)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),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,X3))
            & ( aa_real_real(F,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xd(Diff),X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(X3),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9828_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
                 => ( ( X3 != C )
                   => ? [T4: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,X3),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),X3) ) )
                        & ( aa_real_real(F,X3) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xe(Diff),C),X3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X3),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9829_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),B)
                  & ( aa_real_real(F,B) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xf(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9830_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C)
                  & ( aa_real_real(F,A) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xf(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9831_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K2: nat,B4: real] :
      ( ! [M: nat,T4: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
            & 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,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K2) )
       => ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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_xh(Nb),Diff),B4),M2),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xi(Diff),M2),T5),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))),aa_real_real(times_times_real(B4),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9832_DERIV__arctan__series,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_xj,suminf_real(aTP_Lamm_xk(X3)),topolo2177554685111907308n_real(X3,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9833_DERIV__real__root__generic,axiom,
    ! [Nb: nat,X3: real,D2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( X3 != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
             => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real)
               => ( D2 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D2,topolo2177554685111907308n_real(X3,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9834_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_xl(F),X5)) )
     => ( 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_xn(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_xl(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9835_list__encode_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ~ ! [X5: nat,Xs2: list_nat] : X3 != cons_nat(X5,Xs2) ) ).

% list_encode.cases
tff(fact_9836_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9837_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9838_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(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9839_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_xo,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9840_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9841_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)
         => ! [N7: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_xp(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_xp(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))),N7)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_xp(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))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9842_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))
         => ! [N7: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_xp(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_xp(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))),N7))),groups6591440286371151544t_real(aTP_Lamm_xp(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))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9843_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_xp(A))),groups6591440286371151544t_real(aTP_Lamm_xp(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))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9844_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9845_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9846_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xq(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9847_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X6,N)) )
     => filterlim_nat_real(aTP_Lamm_xr(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9848_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_xs,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9849_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_xt(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9850_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(plus_plus_real(aa_nat_real(F,N7)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9851_LIMSEQ__realpow__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),one_one_real)
       => filterlim_nat_real(power_power_real(X3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9852_LIMSEQ__divide__realpow__zero,axiom,
    ! [X3: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X3)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_xu(X3),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9853_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(power_power_real(aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9854_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(power_power_real(C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9855_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X3)
     => filterlim_nat_real(aTP_Lamm_xv(X3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9856_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_xw(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9857_tendsto__exp__limit__sequentially,axiom,
    ! [X3: real] : filterlim_nat_real(aTP_Lamm_xx(X3),topolo2815343760600316023s_real(aa_real_real(exp_real,X3)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9858_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_xy(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9859_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_xp(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9860_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_xp(A)) ) ) ) ).

% summable
tff(fact_9861_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_xz(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_ya(Theta),K),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9862_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_yb(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_ya(Theta),K),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9863_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_yc(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xp(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9864_zeroseq__arctan__series,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real)
     => filterlim_nat_real(aTP_Lamm_hv(X3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9865_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_yc(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xp(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9866_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_xp(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))),Nb)))),suminf_real(aTP_Lamm_xp(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9867_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_xp(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))),N7)))),L3)
              & filterlim_nat_real(aTP_Lamm_yc(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_xp(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))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_yd(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9868_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_yd(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xp(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9869_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_yd(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xp(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9870_tendsto__power__div__exp__0,axiom,
    ! [K2: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_ye,K2),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9871_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_9872_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_9873_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_9874_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yf(Nb),F),at_top_real,F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9875_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yf(Nb),F),at_bot_real,F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9876_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_9877_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_9878_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_9879_greaterThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1210151606488870762an_nat,K2)),insert_nat(aa_nat_nat(suc,K2),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9880_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yg),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9881_normalize__negative,axiom,
    ! [Q3: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q3),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q3))) ) ) ).

% normalize_negative
tff(fact_9882_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9883_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9884_quotient__of__number_I3_J,axiom,
    ! [K2: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9885_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9886_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9887_quotient__of__number_I5_J,axiom,
    ! [K2: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9888_quotient__of__div,axiom,
    ! [R2: rat,Nb: int,D: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Nb),D) )
     => ( R2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_int_rat(ring_1_of_int_rat,Nb)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9889_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yi(Q3)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9890_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yk(Q3)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9891_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ym(Q3)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9892_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yo(Q3)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9893_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q3: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% quotient_of_denom_pos
tff(fact_9894_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yp),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9895_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yq),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9896_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% normalize_denom_pos
tff(fact_9897_normalize__crossproduct,axiom,
    ! [Q3: int,Sb: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( Sb != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),Sb)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),Sb) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9898_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9899_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_9900_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_yr,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9901_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_ys(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9902_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_yt(P),K2),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9903_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_ch,at_top_nat) ).

% eventually_False_sequentially
tff(fact_9904_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N8: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N3)
         => aa_nat_o(P,N3) ) ) ).

% eventually_sequentially
tff(fact_9905_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X5)
         => aa_nat_o(P,X5) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9906_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F4) ) ).

% le_sequentially
tff(fact_9907_sequentially__offset,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_yt(P),K2),at_top_nat) ) ).

% sequentially_offset
tff(fact_9908_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X6) = $ite(X6 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X6)) ).

% Sup_nat_def
tff(fact_9909_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2))) ) ).

% card_le_Suc_Max
tff(fact_9910_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yu(Mb),Nb)))) ).

% divide_nat_def
tff(fact_9911_Frct__code__post_I5_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,K2)) ).

% Frct_code_post(5)
tff(fact_9912_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P2)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P2)),product_snd_int_int(P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P2)),a)) ),
        $ite(
          product_snd_int_int(P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P2)),product_snd_int_int(P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9913_gcd__neg__numeral__2__int,axiom,
    ! [X3: int,Nb: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X3),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X3),aa_num_int(numeral_numeral_int,Nb)) ).

% gcd_neg_numeral_2_int
tff(fact_9914_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,X3: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),X3) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,Nb)),X3) ).

% gcd_neg_numeral_1_int
tff(fact_9915_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_9916_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_9917_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,A)),B)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(7)
tff(fact_9918_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),aa_int_int(uminus_uminus_int,B))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(8)
tff(fact_9919_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9920_Frct__code__post_I4_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K2) ).

% Frct_code_post(4)
tff(fact_9921_Frct__code__post_I6_J,axiom,
    ! [K2: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_num_rat(numeral_numeral_rat,K2)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9922_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X3),Xa2)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : X3 = vEBT_Leaf((Uu),(Uv))
         => ( (Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(Mima,Deg,TreeList,Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg = Xa2 )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary2),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Deg),TreeList),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9923_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9924_gcd__le1__nat,axiom,
    ! [A: nat,B: 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),B)),A) ) ).

% gcd_le1_nat
tff(fact_9925_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9926_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9927_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9928_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9929_Gcd__in,axiom,
    ! [A4: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_set_nat_o(member_nat(A3),A4)
         => ( aa_set_nat_o(member_nat(B3),A4)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A4) ) )
     => ( ( A4 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A4)),A4) ) ) ).

% Gcd_in
tff(fact_9930_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9931_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9932_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9933_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9934_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X5: nat,Y3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X5))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X5)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X5))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X5)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9935_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(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9936_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9937_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg2: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg2)
    <=> ( ( Dega = Deg2 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X4),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X9)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9938_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X3),Xa2)
     => ( ( ? [Uu: $o,Uv: $o] : X3 = vEBT_Leaf((Uu),(Uv))
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(Mima,Deg,TreeList,Summary2) )
             => ( ( Deg = Xa2 )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary2),m2)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Deg),TreeList),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9939_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X3),Xa2)
     => ( ( ? [Uu: $o,Uv: $o] : X3 = vEBT_Leaf((Uu),(Uv))
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(Mima,Deg,TreeList,Summary2) )
             => ~ ( ( Deg = Xa2 )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary2),m2)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Deg),TreeList),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9940_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Mima,Deg,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary2),Xa2))
                 => ( ( Deg = Xa2 )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary2),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Deg),TreeList),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9941_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X3),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Mima,Deg,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary2),Xa2))
                 => ~ ( ( Deg = Xa2 )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary2),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Deg),TreeList),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9942_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X3),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X3,Xa2))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3 = vEBT_Leaf((Uu),(Uv)) )
             => ( ( (Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Mima,Deg,TreeList,Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg = Xa2 )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary2),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yv(Deg),TreeList),Summary2),n),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9943_gcd__nat_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X3),Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero_nat,X3,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa2))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9944_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K2) ).

% atLeast_Suc_greaterThan
tff(fact_9945_atLeast__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atLeast_nat,K2)),insert_nat(K2,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9946_upto_Opelims,axiom,
    ! [X3: int,Xa2: int,Y: list_int] :
      ( ( upto(X3,Xa2) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa2),cons_int(X3,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int),Xa2)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_9947_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_9948_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9949_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9950_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9951_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9952_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9953_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% less_eq
tff(fact_9954_Bseq__realpow,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => bfun_nat_real(power_power_real(X3),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9955_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_xa,Nb)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9956_inj__Suc,axiom,
    ! [N4: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N4) ).

% inj_Suc
tff(fact_9957_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K2: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wq,K2)),N4) ) ).

% inj_on_diff_nat
tff(fact_9958_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9959_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_9960_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% pred_nat_trancl_eq_le
tff(fact_9961_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_9962_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_9963_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_Suc_Suc
tff(fact_9964_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_9965_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9966_min__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% min_numeral_Suc
tff(fact_9967_min__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% min_Suc_numeral
tff(fact_9968_min__diff,axiom,
    ! [Mb: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),I) ).

% min_diff
tff(fact_9969_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9970_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).

% nat_mult_min_right
tff(fact_9971_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_min_left
tff(fact_9972_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_yw,Nb)),Mb) ).

% min_Suc1
tff(fact_9973_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_yx,Nb)),Mb) ).

% min_Suc2
tff(fact_9974_drop__upt,axiom,
    ! [Mb: nat,I: nat,J: nat] : drop_nat(Mb,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb),J) ).

% drop_upt
tff(fact_9975_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9976_take__upt,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)),Nb)
     => ( take_nat(Mb,upt(I,Nb)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)) ) ) ).

% take_upt
tff(fact_9977_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_9978_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_9979_nth__upt,axiom,
    ! [I: nat,K2: 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),K2)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) ) ) ).

% nth_upt
tff(fact_9980_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_9981_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_9982_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q3) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9983_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_9984_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_9985_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yy,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% map_add_upt
tff(fact_9986_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_9987_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_9988_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_9989_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_yz,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_9990_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K2: 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),K2)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_9991_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X3: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X3,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = X3 )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9992_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_9993_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_9994_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_9995_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mg),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) ) ) ).

% sum_list_upt
tff(fact_9996_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_za(Mb),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_zb(Mb),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_zc(Mb),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9997_card__length__sum__list,axiom,
    ! [Mb: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_za(Mb),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),Mb)),one_one_nat)),N4) ).

% card_length_sum_list
tff(fact_9998_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_9999_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_10000_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10001_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10002_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zd,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_10003_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_10004_prod__encode__eq,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,X3) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Y) )
    <=> ( X3 = Y ) ) ).

% prod_encode_eq
tff(fact_10005_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_10006_bij__prod__encode,axiom,
    aa_set_nat_o(bij_be5333170631980326235at_nat(nat_prod_encode,top_to4669805908274784177at_nat),top_top_set_nat) ).

% bij_prod_encode
tff(fact_10007_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(nat_prod_encode),A4) ).

% inj_prod_encode
tff(fact_10008_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_2
tff(fact_10009_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_1
tff(fact_10010_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_ze) ).

% prod_encode_def
tff(fact_10011_list__encode_Oelims,axiom,
    ! [X3: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X3) = Y )
     => ( ( ( X3 = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs2: list_nat] :
              ( ( X3 = cons_nat(X5,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_10012_inj__list__encode,axiom,
    ! [A4: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A4) ).

% inj_list_encode
tff(fact_10013_list__encode__eq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X3) = aa_list_nat_nat(nat_list_encode,Y) )
    <=> ( X3 = Y ) ) ).

% list_encode_eq
tff(fact_10014_bij__list__encode,axiom,
    aa_set_nat_o(bij_be8532844293280997160at_nat(nat_list_encode,top_top_set_list_nat),top_top_set_nat) ).

% bij_list_encode
tff(fact_10015_surj__list__encode,axiom,
    image_list_nat_nat(nat_list_encode,top_top_set_list_nat) = top_top_set_nat ).

% surj_list_encode
tff(fact_10016_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_10017_list__encode_Osimps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(X3,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_10018_list__encode_Opelims,axiom,
    ! [X3: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X3) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),X3)
       => ( ( ( X3 = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X5: nat,Xs2: list_nat] :
                ( ( X3 = cons_nat(X5,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X5,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_10019_and__not__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( bit_and_not_num(X3,Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(X3,Xa2))
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X3 = one )
             => ! [N: num] :
                  ( ( Xa2 = aa_num_num(bit0,N) )
                 => ( ( Y = some_num(one) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
           => ( ( ( X3 = one )
               => ! [N: num] :
                    ( ( Xa2 = aa_num_num(bit1,N) )
                   => ( ( Y = none_num )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
             => ( ! [M: num] :
                    ( ( X3 = aa_num_num(bit0,M) )
                   => ( ( Xa2 = one )
                     => ( ( Y = some_num(aa_num_num(bit0,M)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M),one)) ) ) )
               => ( ! [M: num] :
                      ( ( X3 = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xa2 = aa_num_num(bit0,N) )
                         => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M,N)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))) ) ) )
                 => ( ! [M: num] :
                        ( ( X3 = aa_num_num(bit0,M) )
                       => ! [N: num] :
                            ( ( Xa2 = aa_num_num(bit1,N) )
                           => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M,N)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))) ) ) )
                   => ( ! [M: num] :
                          ( ( X3 = aa_num_num(bit1,M) )
                         => ( ( Xa2 = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M),one)) ) ) )
                     => ( ! [M: num] :
                            ( ( X3 = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit0,N) )
                               => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_wt,bit_and_not_num(M,N)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))) ) ) )
                       => ~ ! [M: num] :
                              ( ( X3 = aa_num_num(bit1,M) )
                             => ! [N: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N) )
                                 => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M,N)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_10020_and__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X3,Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(X3,Xa2))
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = some_num(one) )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X3 = one )
             => ! [N: num] :
                  ( ( Xa2 = aa_num_num(bit0,N) )
                 => ( ( Y = none_num )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
           => ( ( ( X3 = one )
               => ! [N: num] :
                    ( ( Xa2 = aa_num_num(bit1,N) )
                   => ( ( Y = some_num(one) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
             => ( ! [M: num] :
                    ( ( X3 = aa_num_num(bit0,M) )
                   => ( ( Xa2 = one )
                     => ( ( Y = none_num )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M),one)) ) ) )
               => ( ! [M: num] :
                      ( ( X3 = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xa2 = aa_num_num(bit0,N) )
                         => ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))) ) ) )
                 => ( ! [M: num] :
                        ( ( X3 = aa_num_num(bit0,M) )
                       => ! [N: num] :
                            ( ( Xa2 = aa_num_num(bit1,N) )
                           => ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))) ) ) )
                   => ( ! [M: num] :
                          ( ( X3 = aa_num_num(bit1,M) )
                         => ( ( Xa2 = one )
                           => ( ( Y = some_num(one) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M),one)) ) ) )
                     => ( ! [M: num] :
                            ( ( X3 = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit0,N) )
                               => ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M,N)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))) ) ) )
                       => ~ ! [M: num] :
                              ( ( X3 = aa_num_num(bit1,M) )
                             => ! [N: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N) )
                                 => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_wt,bit_un7362597486090784418nd_num(M,N)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_10021_xor__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( bit_un2480387367778600638or_num(X3,Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(X3,Xa2))
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X3 = one )
             => ! [N: num] :
                  ( ( Xa2 = aa_num_num(bit0,N) )
                 => ( ( Y = some_num(aa_num_num(bit1,N)) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
           => ( ( ( X3 = one )
               => ! [N: num] :
                    ( ( Xa2 = aa_num_num(bit1,N) )
                   => ( ( Y = some_num(aa_num_num(bit0,N)) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
             => ( ! [M: num] :
                    ( ( X3 = aa_num_num(bit0,M) )
                   => ( ( Xa2 = one )
                     => ( ( Y = some_num(aa_num_num(bit1,M)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M),one)) ) ) )
               => ( ! [M: num] :
                      ( ( X3 = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xa2 = aa_num_num(bit0,N) )
                         => ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))) ) ) )
                 => ( ! [M: num] :
                        ( ( X3 = aa_num_num(bit0,M) )
                       => ! [N: num] :
                            ( ( Xa2 = aa_num_num(bit1,N) )
                           => ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N))) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))) ) ) )
                   => ( ! [M: num] :
                          ( ( X3 = aa_num_num(bit1,M) )
                         => ( ( Xa2 = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M),one)) ) ) )
                     => ( ! [M: num] :
                            ( ( X3 = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit0,N) )
                               => ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N))) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))) ) ) )
                       => ~ ! [M: num] :
                              ( ( X3 = aa_num_num(bit1,M) )
                             => ! [N: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N) )
                                 => ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M,N)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_10022_or__not__num__neg_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X3),Xa2) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(X3,Xa2))
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = one )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X3 = one )
             => ! [M: num] :
                  ( ( Xa2 = aa_num_num(bit0,M) )
                 => ( ( Y = aa_num_num(bit1,M) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit0,M))) ) ) )
           => ( ( ( X3 = one )
               => ! [M: num] :
                    ( ( Xa2 = aa_num_num(bit1,M) )
                   => ( ( Y = aa_num_num(bit1,M) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit1,M))) ) ) )
             => ( ! [N: num] :
                    ( ( X3 = aa_num_num(bit0,N) )
                   => ( ( Xa2 = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),one)) ) ) )
               => ( ! [N: num] :
                      ( ( X3 = aa_num_num(bit0,N) )
                     => ! [M: num] :
                          ( ( Xa2 = aa_num_num(bit0,M) )
                         => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit0,M))) ) ) )
                 => ( ! [N: num] :
                        ( ( X3 = aa_num_num(bit0,N) )
                       => ! [M: num] :
                            ( ( Xa2 = aa_num_num(bit1,M) )
                           => ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit1,M))) ) ) )
                   => ( ! [N: num] :
                          ( ( X3 = aa_num_num(bit1,N) )
                         => ( ( Xa2 = one )
                           => ( ( Y = one )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),one)) ) ) )
                     => ( ! [N: num] :
                            ( ( X3 = aa_num_num(bit1,N) )
                           => ! [M: num] :
                                ( ( Xa2 = aa_num_num(bit0,M) )
                               => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit0,M))) ) ) )
                       => ~ ! [N: num] :
                              ( ( X3 = aa_num_num(bit1,N) )
                             => ! [M: num] :
                                  ( ( Xa2 = aa_num_num(bit1,M) )
                                 => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit1,M))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_10023_Field__natLeq__on,axiom,
    ! [Nb: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Nb)) ).

% Field_natLeq_on
tff(fact_10024_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(F,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_10025_DeMoivre2,axiom,
    ! [R2: real,A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(rcis(R2),A)),Nb) = aa_real_complex(rcis(aa_nat_real(power_power_real(R2),Nb)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre2
tff(fact_10026_divmod__nat__code,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $let(
        k: code_integer,
        k:= aa_nat_Code_integer(code_integer_of_nat,Mb),
        $let(
          l: code_integer,
          l:= aa_nat_Code_integer(code_integer_of_nat,Nb),
          produc8678311845419106900er_nat(code_nat_of_integer,code_nat_of_integer,
            $ite(
              k = zero_z3403309356797280102nteger,
              produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
              $ite(l = zero_z3403309356797280102nteger,produc1086072967326762835nteger(zero_z3403309356797280102nteger,k),code_divmod_abs(k,l)) )) ) ) ).

% divmod_nat_code
tff(fact_10027_integer__of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Code_integer(code_integer_of_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% integer_of_nat_numeral
tff(fact_10028_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M4)
         => ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N3)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M4)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10029_le__enumerate,axiom,
    ! [S2: set_nat,Nb: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S2),Nb)) ) ).

% le_enumerate
tff(fact_10030_finite__le__enumerate,axiom,
    ! [S2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_set_nat_nat(finite_card_nat,S2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S2),Nb)) ) ) ).

% finite_le_enumerate
tff(fact_10031_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_rz(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aTP_Lamm_zg(Mb)) ).

% pairs_le_eq_Sigma
tff(fact_10032_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_10033_Least__Suc2,axiom,
    ! [P: fun_nat_o,Nb: nat,Q: fun_nat_o,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(Q,Mb)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K))
              <=> aa_nat_o(Q,K) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_10034_Least__Suc,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_ys(P))) ) ) ) ).

% Least_Suc
tff(fact_10035_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A4)) = vimage_nat_nat(suc,A4) ).

% vimage_Suc_insert_0
tff(fact_10036_finite__vimage__Suc__iff,axiom,
    ! [F4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F4))
    <=> aa_set_nat_o(finite_finite_nat,F4) ) ).

% finite_vimage_Suc_iff
tff(fact_10037_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A4: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Nb),A4)) = insert_nat(Nb,vimage_nat_nat(suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_10038_set__decode__div__2,axiom,
    ! [X3: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,X3)) ).

% set_decode_div_2
tff(fact_10039_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A4)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_10040_Restr__natLeq,axiom,
    ! [Nb: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Nb)),aTP_Lamm_zh(Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb))) ).

% Restr_natLeq
tff(fact_10041_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_10042_Restr__natLeq2,axiom,
    ! [Nb: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Nb),aTP_Lamm_zi(Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb))) ).

% Restr_natLeq2
tff(fact_10043_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).

% tl_upt
tff(fact_10044_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_10045_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_10046_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Nb))) ).

% sqr.simps(2)
tff(fact_10047_pow_Osimps_I2_J,axiom,
    ! [X3: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X3),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,X3),Y)) ).

% pow.simps(2)
tff(fact_10048_Rep__unit__induct,axiom,
    ! [Y: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ( ! [X5: product_unit] : aa_o_o(P,aa_Product_unit_o(product_Rep_unit,X5))
       => aa_o_o(P,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_10049_Abs__unit__inject,axiom,
    ! [X3: $o,Y: $o] :
      ( aa_set_o_o(member_o((X3)),insert_o($true,bot_bot_set_o))
     => ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
       => ( ( aa_o_Product_unit(product_Abs_unit,(X3)) = aa_o_Product_unit(product_Abs_unit,(Y)) )
        <=> ( (X3)
          <=> (Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_10050_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ( aa_Product_unit_o(product_Rep_unit,aa_o_Product_unit(product_Abs_unit,(Y)))
      <=> (Y) ) ) ).

% Abs_unit_inverse
tff(fact_10051_Rep__unit__inject,axiom,
    ! [X3: product_unit,Y: product_unit] :
      ( ( aa_Product_unit_o(product_Rep_unit,X3)
      <=> aa_Product_unit_o(product_Rep_unit,Y) )
    <=> ( X3 = Y ) ) ).

% Rep_unit_inject
tff(fact_10052_Rep__unit__inverse,axiom,
    ! [X3: product_unit] : aa_o_Product_unit(product_Abs_unit,aa_Product_unit_o(product_Rep_unit,X3)) = X3 ).

% Rep_unit_inverse
tff(fact_10053_Rep__unit,axiom,
    ! [X3: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(product_Rep_unit,X3)),insert_o($true,bot_bot_set_o)) ).

% Rep_unit
tff(fact_10054_Abs__unit__cases,axiom,
    ! [X3: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X3 = aa_o_Product_unit(product_Abs_unit,(Y3)) )
         => ~ aa_set_o_o(member_o((Y3)),insert_o($true,bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_10055_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ~ ! [X5: product_unit] :
            ( (Y)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X5) ) ) ).

% Rep_unit_cases
tff(fact_10056_Abs__unit__induct,axiom,
    ! [P: fun_Product_unit_o,X3: product_unit] :
      ( ! [Y3: $o] :
          ( aa_set_o_o(member_o((Y3)),insert_o($true,bot_bot_set_o))
         => aa_Product_unit_o(P,aa_o_Product_unit(product_Abs_unit,(Y3))) )
     => aa_Product_unit_o(P,X3) ) ).

% Abs_unit_induct
tff(fact_10057_type__definition__unit,axiom,
    aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),insert_o($true,bot_bot_set_o)) ).

% type_definition_unit
tff(fact_10058_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_10059_natLeq__on__wo__rel,axiom,
    ! [Nb: nat] : aa_set3702741264012733747_nat_o(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb)))) ).

% natLeq_on_wo_rel
tff(fact_10060_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_10061_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa2),Xa2)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),X3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X3)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa2)),product_snd_int_int(X3))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X3)),product_snd_int_int(Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X3)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_10062_one__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat.rsp
tff(fact_10063_zero__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat.rsp
tff(fact_10064_uminus__rat_Oabs__eq,axiom,
    ! [X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),X3)
     => ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X3)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,product_fst_int_int(X3))),product_snd_int_int(X3))) ) ) ).

% uminus_rat.abs_eq
tff(fact_10065_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa2),Xa2)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),X3)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X3)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa2)),product_fst_int_int(X3))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X3)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_10066_inverse__rat_Oabs__eq,axiom,
    ! [X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),X3)
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X3)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
            $ite(product_fst_int_int(X3) = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(X3)),product_fst_int_int(X3)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_10067_inverse__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_zj),aTP_Lamm_zj) ).

% inverse_rat.rsp
tff(fact_10068_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M6: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),M6),at_top_nat)
     => ( ! [M: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),M)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,M)),aa_nat_real(F,N)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_10069_Bseq__mono__convergent,axiom,
    ! [X6: fun_nat_real] :
      ( bfun_nat_real(X6,at_top_nat)
     => ( ! [M: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M)),aa_nat_real(X6,N)) )
       => topolo7531315842566124627t_real(X6) ) ) ).

% Bseq_mono_convergent
tff(fact_10070_convergent__realpow,axiom,
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),one_one_real)
       => topolo7531315842566124627t_real(power_power_real(X3)) ) ) ).

% convergent_realpow
tff(fact_10071_uminus__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_zk),aTP_Lamm_zk) ).

% uminus_rat.rsp
tff(fact_10072_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M6: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),M6),at_top_nat)
     => ( ! [M: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),M)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_10073_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_zl,aTP_Lamm_zl) ).

% plus_rat.rsp
tff(fact_10074_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_10075_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_10076_Suc_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc) ).

% Suc.rsp
tff(fact_10077_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_zm,aTP_Lamm_zm) ).

% sub.rsp
tff(fact_10078_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_zn,aTP_Lamm_zn) ).

% Fract.rsp
tff(fact_10079_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_zo,aTP_Lamm_zo) ).

% times_rat.rsp
tff(fact_10080_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_zl,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_10081_inverse__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_zj),inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_10082_one__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),one_one_rat) ).

% one_rat.transfer
tff(fact_10083_zero__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),zero_zero_rat) ).

% zero_rat.transfer
tff(fact_10084_uminus__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_zk),uminus_uminus_rat) ).

% uminus_rat.transfer
tff(fact_10085_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_zo,times_times_rat) ).

% times_rat.transfer
tff(fact_10086_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_we),times_times_int) ).

% times_int.transfer
tff(fact_10087_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_wn),minus_minus_int) ).

% minus_int.transfer
tff(fact_10088_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_10089_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_zp,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_10090_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_wf)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_10091_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_10092_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_wh),ord_less_int) ).

% less_int.transfer
tff(fact_10093_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_wj),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_10094_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_wl),plus_plus_int) ).

% plus_int.transfer
tff(fact_10095_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_we),produc4528391901023682385at_nat(aTP_Lamm_we)) ).

% times_int.rsp
tff(fact_10096_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_wn),produc4528391901023682385at_nat(aTP_Lamm_wn)) ).

% minus_int.rsp
tff(fact_10097_intrel__iff,axiom,
    ! [X3: nat,Y: nat,U: nat,V2: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),V2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_10098_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_10099_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_wf)),produc2626176000494625587at_nat(aTP_Lamm_wf)) ).

% uminus_int.rsp
tff(fact_10100_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_10101_intrel__def,axiom,
    intrel = produc6395135316500328296_nat_o(aTP_Lamm_zr) ).

% intrel_def
tff(fact_10102_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_wh),produc6395135316500328296_nat_o(aTP_Lamm_wh)) ).

% less_int.rsp
tff(fact_10103_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_wj),produc6395135316500328296_nat_o(aTP_Lamm_wj)) ).

% less_eq_int.rsp
tff(fact_10104_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_wl),produc4528391901023682385at_nat(aTP_Lamm_wl)) ).

% plus_int.rsp
tff(fact_10105_pair__lessI2,axiom,
    ! [A: nat,B: nat,Sb: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Sb),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Sb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_10106_pair__less__iff1,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ).

% pair_less_iff1
tff(fact_10107_pair__lessI1,axiom,
    ! [A: nat,B: nat,Sb: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Sb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_10108_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_eq,aTP_Lamm_mb),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_10109_pair__leqI2,axiom,
    ! [A: nat,B: nat,Sb: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Sb),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Sb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_10110_pair__leqI1,axiom,
    ! [A: nat,B: nat,Sb: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Sb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_10111_wmin__insertI,axiom,
    ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_10112_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_max_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X3,XS),YS)),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_10113_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X6)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6)),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_10114_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_weak) ).

% wmin_emptyI
tff(fact_10115_min__weak__def,axiom,
    fun_min_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,min_ex6901939911449802026at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_10116_max__weak__def,axiom,
    fun_max_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,max_ex8135407076693332796at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_10117_smin__insertI,axiom,
    ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_10118_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Y6)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y6)),fun_max_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X3,X6),Y6)),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_10119_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_strict) ) ).

% smin_emptyI
tff(fact_10120_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y6)
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y6)),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_10121_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_10122_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_10123_rat__number__expand_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int) ).

% rat_number_expand(5)
tff(fact_10124_normalize__eq,axiom,
    ! [A: int,B: int,P2: int,Q3: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P2),Q3) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% normalize_eq
tff(fact_10125_quotient__of__eq,axiom,
    ! [A: int,B: int,P2: int,Q3: int] :
      ( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P2),Q3) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% quotient_of_eq
tff(fact_10126_rat__number__expand_I3_J,axiom,
    ! [K2: num] : aa_num_rat(numeral_numeral_rat,K2) = aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,K2)),one_one_int) ).

% rat_number_expand(3)
tff(fact_10127_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,W)),one_one_int) = aa_num_rat(numeral_numeral_rat,W) ).

% rat_number_collapse(3)
tff(fact_10128_quotient__of__Fract,axiom,
    ! [A: int,B: int] : quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ).

% quotient_of_Fract
tff(fact_10129_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X3: int] :
      aa_int_rat(aa_int_fun_int_rat(fract,Xa2),X3) = aa_Pro6166618844887768661nt_rat(abs_Rat,
        $ite(X3 = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa2),X3))) ).

% Fract.abs_eq
tff(fact_10130_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_zn,fract) ).

% Fract.transfer
tff(fact_10131_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) ).

% rat_number_collapse(4)
tff(fact_10132_less__eq__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Mb,Nb)
    <=> extended_case_enat_o(aTP_Lamm_zs(Mb),$true,Nb) ) ).

% less_eq_enat_def
tff(fact_10133_less__than__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y)),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y) ) ).

% less_than_iff
tff(fact_10134_natLeq__on__well__order__on,axiom,
    ! [Nb: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Nb))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb)))) ).

% natLeq_on_well_order_on
tff(fact_10135_natLeq__on__Well__order,axiom,
    ! [Nb: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb))))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_zf(Nb)))) ).

% natLeq_on_Well_order
tff(fact_10136_prod__decode__triangle__add,axiom,
    ! [K2: nat,Mb: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),Mb)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),Mb) ).

% prod_decode_triangle_add
tff(fact_10137_prod__decode__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode,X3) = aa_nat7603221163087113075at_nat(nat_prod_decode,Y) )
    <=> ( X3 = Y ) ) ).

% prod_decode_eq
tff(fact_10138_prod__decode__inverse,axiom,
    ! [Nb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode,Nb)) = Nb ).

% prod_decode_inverse
tff(fact_10139_prod__encode__inverse,axiom,
    ! [X3: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_Pro4243173770044397973at_nat(nat_prod_encode,X3)) = X3 ).

% prod_encode_inverse
tff(fact_10140_inj__prod__decode,axiom,
    ! [A4: set_nat] : aa_set_nat_o(inj_on5538052773655684606at_nat(nat_prod_decode),A4) ).

% inj_prod_decode
tff(fact_10141_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_10142_bij__prod__decode,axiom,
    aa_set3702741264012733747_nat_o(bij_be8693218025023041337at_nat(nat_prod_decode,top_top_set_nat),top_to4669805908274784177at_nat) ).

% bij_prod_decode
tff(fact_10143_surj__prod__decode,axiom,
    image_5846123807819985514at_nat(nat_prod_decode,top_top_set_nat) = top_to4669805908274784177at_nat ).

% surj_prod_decode
tff(fact_10144_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_o] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),A0)
     => ( ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
         => aa_nat_o(P,zero_zero_nat) )
       => ( ! [N: nat] :
              ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N))
             => ( ! [X: nat,Y5: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y5) = aa_nat7603221163087113075at_nat(nat_prod_decode,N) )
                   => aa_nat_o(P,Y5) )
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_10145_list__decode_Oelims,axiom,
    ! [X3: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N: nat] :
              ( ( X3 = aa_nat_nat(suc,N) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_zt,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ) ) ).

% list_decode.elims
tff(fact_10146_list__encode__inverse,axiom,
    ! [X3: list_nat] : aa_nat_list_nat(nat_list_decode,aa_list_nat_nat(nat_list_encode,X3)) = X3 ).

% list_encode_inverse
tff(fact_10147_list__decode__inverse,axiom,
    ! [Nb: nat] : aa_list_nat_nat(nat_list_encode,aa_nat_list_nat(nat_list_decode,Nb)) = Nb ).

% list_decode_inverse
tff(fact_10148_list__decode_Opsimps_I1_J,axiom,
    ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
   => ( aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_10149_inj__list__decode,axiom,
    ! [A4: set_nat] : aa_set_nat_o(inj_on_nat_list_nat(nat_list_decode),A4) ).

% inj_list_decode
tff(fact_10150_list__decode__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X3) = aa_nat_list_nat(nat_list_decode,Y) )
    <=> ( X3 = Y ) ) ).

% list_decode_eq
tff(fact_10151_list__decode_Osimps_I1_J,axiom,
    aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_10152_list__decode_Opsimps_I2_J,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,Nb))
     => ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,Nb)) = produc2761476792215241774st_nat(aTP_Lamm_zt,aa_nat7603221163087113075at_nat(nat_prod_decode,Nb)) ) ) ).

% list_decode.psimps(2)
tff(fact_10153_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat(nat_list_decode,top_top_set_nat,top_top_set_list_nat) ).

% bij_list_decode
tff(fact_10154_surj__list__decode,axiom,
    image_nat_list_nat(nat_list_decode,top_top_set_nat) = top_top_set_list_nat ).

% surj_list_decode
tff(fact_10155_list__decode_Osimps_I2_J,axiom,
    ! [Nb: nat] : aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,Nb)) = produc2761476792215241774st_nat(aTP_Lamm_zt,aa_nat7603221163087113075at_nat(nat_prod_decode,Nb)) ).

% list_decode.simps(2)
tff(fact_10156_list__decode_Opelims,axiom,
    ! [X3: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X3) = Y )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),X3)
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( X3 = aa_nat_nat(suc,N) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_zt,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_10157_compute__powr__real,axiom,
    ! [B: real,I: real] :
      aa_real_real(powr_real2(B),I) = $ite(
        aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real),
        abort_real(literal2($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_zu(B),I)),
        $ite(
          aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I,
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),I),aa_nat_real(power_power_real(B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I)))))),
          abort_real(literal2($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_zu(B),I)) ) ) ).

% compute_powr_real
tff(fact_10158_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_zj) ).

% inverse_rat_def
tff(fact_10159_card__of__nat,axiom,
    aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).

% card_of_nat
tff(fact_10160_card__of__Field__natLeq,axiom,
    aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).

% card_of_Field_natLeq
tff(fact_10161_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_zk) ).

% uminus_rat_def
tff(fact_10162_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_zl) ).

% plus_rat_def
tff(fact_10163_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_zo) ).

% times_rat_def
tff(fact_10164_ctwo__Cnotzero,axiom,
    ( ~ member444158400953824016od_o_o(produc763777882069021527od_o_o(bNF_Cardinal_ctwo,bNF_Cardinal_czero_o),bNF_We2654380646378065620so_o_o)
    & bNF_Ca8331644756375544342r_on_o(field_o(bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_10165_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat(produc8517790099723286449at_nat(bNF_Cardinal_ctwo,bNF_Ca8665028551170535155natLeq),bNF_We8182288985678559134_o_nat) ).

% ctwo_ordLess_natLeq
tff(fact_10166_numeral__le__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% numeral_le_enat_iff
tff(fact_10167_plus__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : plus_p3455044024723400733d_enat(extended_enat2(Mb),extended_enat2(Nb)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% plus_enat_simps(1)
tff(fact_10168_enat__ord__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ord_le2932123472753598470d_enat(extended_enat2(Mb),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% enat_ord_simps(1)
tff(fact_10169_numeral__less__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% numeral_less_enat_iff
tff(fact_10170_iadd__le__enat__iff,axiom,
    ! [X3: extended_enat,Y: extended_enat,Nb: nat] :
      ( ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(X3,Y),extended_enat2(Nb))
    <=> ? [Y7: nat,X10: nat] :
          ( ( X3 = extended_enat2(X10) )
          & ( Y = extended_enat2(Y7) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X10),Y7)),Nb) ) ) ).

% iadd_le_enat_iff
tff(fact_10171_numeral__eq__enat,axiom,
    ! [K2: num] : numera1916890842035813515d_enat(K2) = extended_enat2(aa_num_nat(numeral_numeral_nat,K2)) ).

% numeral_eq_enat
tff(fact_10172_Suc__ile__eq,axiom,
    ! [Mb: nat,Nb: extended_enat] :
      ( ord_le2932123472753598470d_enat(extended_enat2(aa_nat_nat(suc,Mb)),Nb)
    <=> ord_le72135733267957522d_enat(extended_enat2(Mb),Nb) ) ).

% Suc_ile_eq
tff(fact_10173_elimnum,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),Nb)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Infoa,Dega,TreeLista,Summarya),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = vEBT_Node(Infoa,Dega,TreeLista,Summarya) ) ) ).

% elimnum
tff(fact_10174_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Infoa,Dega,TreeLista,Summarya),extended_enat2(L)) = vEBT_Node(Infoa,Dega,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_zv(Dega),TreeLista)),vEBT_VEBT_elim_dead(Summarya,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_10175_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Uu2: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf((A),(B)),Uu2) = vEBT_Leaf((A),(B)) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_10176_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X3,Xa2) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3 = vEBT_Leaf((A3),(B3)) )
           => ( Y != vEBT_Leaf((A3),(B3)) ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = vEBT_Node(Info,Deg,TreeList,Summary2) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y != vEBT_Node(Info,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_zv(Deg),TreeList),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Info,Deg,TreeList,Summary2) )
               => ! [L3: nat] :
                    ( ( Xa2 = extended_enat2(L3) )
                   => ( Y != vEBT_Node(Info,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_zv(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_10177_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Infoa,Dega,TreeLista,Summarya),extend5688581933313929465d_enat) = vEBT_Node(Infoa,Dega,map_VE8901447254227204932T_VEBT(aTP_Lamm_zv(Dega),TreeLista),vEBT_VEBT_elim_dead(Summarya,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_10178_elimcomplete,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),Nb)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Infoa,Dega,TreeLista,Summarya),extend5688581933313929465d_enat) = vEBT_Node(Infoa,Dega,TreeLista,Summarya) ) ) ).

% elimcomplete
tff(fact_10179_numeral__ne__infinity,axiom,
    ! [K2: num] : numera1916890842035813515d_enat(K2) != extend5688581933313929465d_enat ).

% numeral_ne_infinity
tff(fact_10180_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X3: produc7272778201969148633d_enat] :
      ( ! [A3: $o,B3: $o,Uu: extended_enat] : X3 != produc581526299967858633d_enat(vEBT_Leaf((A3),(B3)),Uu)
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X3 != produc581526299967858633d_enat(vEBT_Node(Info,Deg,TreeList,Summary2),extend5688581933313929465d_enat)
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,L3: nat] : X3 != produc581526299967858633d_enat(vEBT_Node(Info,Deg,TreeList,Summary2),extended_enat2(L3)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_10181_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X3,Xa2) = Y )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(X3,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X3 = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = vEBT_Leaf((A3),(B3)) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Leaf((A3),(B3)),Xa2)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = vEBT_Node(Info,Deg,TreeList,Summary2) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y = vEBT_Node(Info,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_zv(Deg),TreeList),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info,Deg,TreeList,Summary2),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X3 = vEBT_Node(Info,Deg,TreeList,Summary2) )
                 => ! [L3: nat] :
                      ( ( Xa2 = extended_enat2(L3) )
                     => ( ( Y = vEBT_Node(Info,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_zv(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info,Deg,TreeList,Summary2),extended_enat2(L3))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_10182_plus__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : plus_p3455044024723400733d_enat(Mb,Nb) = extend3600170679010898289d_enat(aTP_Lamm_zx(Nb),extend5688581933313929465d_enat,Mb) ).

% plus_enat_def
tff(fact_10183_eSuc__def,axiom,
    ! [I: extended_enat] : extended_eSuc(I) = extend3600170679010898289d_enat(aTP_Lamm_zy,extend5688581933313929465d_enat,I) ).

% eSuc_def
tff(fact_10184_eSuc__numeral,axiom,
    ! [K2: num] : extended_eSuc(numera1916890842035813515d_enat(K2)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,K2),one)) ).

% eSuc_numeral
tff(fact_10185_enat__eSuc__iff,axiom,
    ! [Y: nat,X3: extended_enat] :
      ( ( extended_enat2(Y) = extended_eSuc(X3) )
    <=> ? [N3: nat] :
          ( ( Y = aa_nat_nat(suc,N3) )
          & ( extended_enat2(N3) = X3 ) ) ) ).

% enat_eSuc_iff
tff(fact_10186_eSuc__enat__iff,axiom,
    ! [X3: extended_enat,Y: nat] :
      ( ( extended_eSuc(X3) = extended_enat2(Y) )
    <=> ? [N3: nat] :
          ( ( Y = aa_nat_nat(suc,N3) )
          & ( X3 = extended_enat2(N3) ) ) ) ).

% eSuc_enat_iff
tff(fact_10187_eSuc__enat,axiom,
    ! [Nb: nat] : extended_eSuc(extended_enat2(Nb)) = extended_enat2(aa_nat_nat(suc,Nb)) ).

% eSuc_enat
tff(fact_10188_Real_Opositive_Orsp,axiom,
    bNF_re7504242029904815401at_o_o(realrel,fequal_o,aTP_Lamm_zz,aTP_Lamm_zz) ).

% Real.positive.rsp
tff(fact_10189_cauchy__def,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M4)
             => ! [N3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),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(X6,M4)),aa_nat_rat(X6,N3)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_10190_cauchyD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( cauchy(X6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K: nat] :
          ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
           => ! [N7: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N7)
               => 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(X6,M2)),aa_nat_rat(X6,N7)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_10191_cauchyI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K4: nat] :
            ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),M)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),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(X6,M)),aa_nat_rat(X6,N)))),R3) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_10192_le__Real,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real2(X6)),real2(Y6))
        <=> ! [R5: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y6,N3)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_10193_cauchy__not__vanishes,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B3)
            & ? [K: nat] :
              ! [N7: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N7)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N7))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_10194_vanishesD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( vanishes(X6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N7)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N7))),R2) ) ) ) ).

% vanishesD
tff(fact_10195_vanishesI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K4: nat] :
            ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N))),R3) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_10196_vanishes__def,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R5) ) ) ) ).

% vanishes_def
tff(fact_10197_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B3)
            & ? [K: nat] :
                ( ! [N7: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X6,N7))) )
                | ! [N7: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),aa_nat_rat(X6,N7)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_10198_not__positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ aa_real_o(positive,real2(X6))
      <=> ! [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
           => ? [K3: nat] :
              ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_10199_positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( aa_real_o(positive,real2(X6))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(X6,N3)) ) ) ) ) ).

% positive_Real
tff(fact_10200_Real_Opositive_Oabs__eq,axiom,
    ! [X3: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,X3),X3)
     => ( aa_real_o(positive,real2(X3))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(X3,N3)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_10201_Real_Opositive_Otransfer,axiom,
    bNF_re3218747500659316912al_o_o(pcr_real,fequal_o,aTP_Lamm_zz,positive) ).

% Real.positive.transfer
tff(fact_10202_Real_Opositive_Orep__eq,axiom,
    ! [X3: real] :
      ( aa_real_o(positive,X3)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X3),N3)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_10203_Real_Opositive__def,axiom,
    positive = map_fu6719675619867407465at_o_o(rep_real,id_o,aTP_Lamm_zz) ).

% Real.positive_def
tff(fact_10204_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_10205_normalize__stable,axiom,
    ! [Q3: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q3)
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) ) ) ) ).

% normalize_stable
tff(fact_10206_ATP_Olambda__1,axiom,
    ! [Uu2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_zj,Uu2) = $ite(product_fst_int_int(Uu2) = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(Uu2)),product_fst_int_int(Uu2))) ).

% ATP.lambda_1
tff(fact_10207_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_ia,Uu2) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uu2)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_2
tff(fact_10208_ATP_Olambda__3,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_oc,Uu2) = aa_nat_real(power_power_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_3
tff(fact_10209_ATP_Olambda__4,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_vn,Uu2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_4
tff(fact_10210_ATP_Olambda__5,axiom,
    ! [Uu2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_zk,Uu2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,product_fst_int_int(Uu2))),product_snd_int_int(Uu2)) ).

% ATP.lambda_5
tff(fact_10211_ATP_Olambda__6,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_on,Uu2) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu2)),aa_nat_real(power_power_real(zero_zero_real),Uu2)) ).

% ATP.lambda_6
tff(fact_10212_ATP_Olambda__7,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_xo,Uu2) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).

% ATP.lambda_7
tff(fact_10213_ATP_Olambda__8,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_yz,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_8
tff(fact_10214_ATP_Olambda__9,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_pf,Uu2) = aa_complex_complex(plus_plus_complex(Uu2),one_one_complex) ).

% ATP.lambda_9
tff(fact_10215_ATP_Olambda__10,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ph,Uu2) = aa_real_real(plus_plus_real(Uu2),one_one_real) ).

% ATP.lambda_10
tff(fact_10216_ATP_Olambda__11,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_pg,Uu2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),one_one_rat) ).

% ATP.lambda_11
tff(fact_10217_ATP_Olambda__12,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_pj,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),one_one_nat) ).

% ATP.lambda_12
tff(fact_10218_ATP_Olambda__13,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_pi,Uu2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),one_one_int) ).

% ATP.lambda_13
tff(fact_10219_ATP_Olambda__14,axiom,
    ! [Uu2: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_zp,Uu2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),zero_zero_nat) ).

% ATP.lambda_14
tff(fact_10220_ATP_Olambda__15,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_xj,Uu2) = suminf_real(aTP_Lamm_hx(Uu2)) ).

% ATP.lambda_15
tff(fact_10221_ATP_Olambda__16,axiom,
    ! [Uu2: nat] : aa_nat_set_nat(aTP_Lamm_yr,Uu2) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fu,Uu2)) ).

% ATP.lambda_16
tff(fact_10222_ATP_Olambda__17,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_xs,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).

% ATP.lambda_17
tff(fact_10223_ATP_Olambda__18,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_wt,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).

% ATP.lambda_18
tff(fact_10224_ATP_Olambda__19,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_ws,Uu2) = some_num(aa_num_num(bit0,Uu2)) ).

% ATP.lambda_19
tff(fact_10225_ATP_Olambda__20,axiom,
    ! [Uu2: int] : aa_int2565445551602696444nt_int(aTP_Lamm_yp,Uu2) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu2)) ).

% ATP.lambda_20
tff(fact_10226_ATP_Olambda__21,axiom,
    ! [Uu2: int] : aa_int2565445551602696444nt_int(aTP_Lamm_yq,Uu2) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu2)) ).

% ATP.lambda_21
tff(fact_10227_ATP_Olambda__22,axiom,
    ! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_hn,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_22
tff(fact_10228_ATP_Olambda__23,axiom,
    ! [Uu2: nat] : aa_nat_Extended_enat(aTP_Lamm_zy,Uu2) = extended_enat2(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_23
tff(fact_10229_ATP_Olambda__24,axiom,
    ! [Uu2: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aTP_Lamm_zz,Uu2)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Uu2,N3)) ) ) ) ).

% ATP.lambda_24
tff(fact_10230_ATP_Olambda__25,axiom,
    ! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_wr,Uu2) = some_num(one) ).

% ATP.lambda_25
tff(fact_10231_ATP_Olambda__26,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_wy(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_ww(Uua),aTP_Lamm_wx(Uua),Uu2) ).

% ATP.lambda_26
tff(fact_10232_ATP_Olambda__27,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_py(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),zero_zero_complex) ).

% ATP.lambda_27
tff(fact_10233_ATP_Olambda__28,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qb,Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),zero_zero_real) ).

% ATP.lambda_28
tff(fact_10234_ATP_Olambda__29,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_pz(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),zero_zero_rat) ).

% ATP.lambda_29
tff(fact_10235_ATP_Olambda__30,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qa,Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),zero_zero_int) ).

% ATP.lambda_30
tff(fact_10236_ATP_Olambda__31,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_oh(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_31
tff(fact_10237_ATP_Olambda__32,axiom,
    ! [Uu2: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_yg,Uu2),Uua) = $ite(Uu2 = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu2)),Uua)),aa_int_int(abs_abs_int,Uu2))) ).

% ATP.lambda_32
tff(fact_10238_ATP_Olambda__33,axiom,
    ! [Uu2: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_zn,Uu2),Uua) = $ite(Uua = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),Uua)) ).

% ATP.lambda_33
tff(fact_10239_ATP_Olambda__34,axiom,
    ! [Uu2: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_zx(Uu2),Uua) = extend3600170679010898289d_enat(aTP_Lamm_zw(Uua),extend5688581933313929465d_enat,Uu2) ).

% ATP.lambda_34
tff(fact_10240_ATP_Olambda__35,axiom,
    ! [Uu2: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zs(Uu2),Uua)
    <=> extended_case_enat_o(aa_nat_fun_nat_o(aTP_Lamm_eq,Uua),$false,Uu2) ) ).

% ATP.lambda_35
tff(fact_10241_ATP_Olambda__36,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_wz,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_wy(Uua),Uu2) ).

% ATP.lambda_36
tff(fact_10242_ATP_Olambda__37,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ww(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ws,bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_37
tff(fact_10243_ATP_Olambda__38,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_wu(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ws,bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_38
tff(fact_10244_ATP_Olambda__39,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sd(Uu2),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sc(Uu2),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_39
tff(fact_10245_ATP_Olambda__40,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_sb(Uu2),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sa(Uu2),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_40
tff(fact_10246_ATP_Olambda__41,axiom,
    ! [Uu2: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_zl,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu2)),product_snd_int_int(Uua))) ).

% ATP.lambda_41
tff(fact_10247_ATP_Olambda__42,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_hw(Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_42
tff(fact_10248_ATP_Olambda__43,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_ov(Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(power_power_real(Uu2),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_43
tff(fact_10249_ATP_Olambda__44,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_pd(Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(power_power_real(Uu2),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_44
tff(fact_10250_ATP_Olambda__45,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_up(Uu2),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_45
tff(fact_10251_ATP_Olambda__46,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_uo(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_46
tff(fact_10252_ATP_Olambda__47,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_us,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_47
tff(fact_10253_ATP_Olambda__48,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_uq(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_48
tff(fact_10254_ATP_Olambda__49,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ur,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_49
tff(fact_10255_ATP_Olambda__50,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ut(Uu2),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_50
tff(fact_10256_ATP_Olambda__51,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uv,Uu2),Uua) = aa_real_real(divide_divide_real(aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uua)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_51
tff(fact_10257_ATP_Olambda__52,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_uu(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uua)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_52
tff(fact_10258_ATP_Olambda__53,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vm,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_53
tff(fact_10259_ATP_Olambda__54,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vl,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_54
tff(fact_10260_ATP_Olambda__55,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_vz,Uu2),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(Uu2)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_55
tff(fact_10261_ATP_Olambda__56,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hq,Uu2),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).

% ATP.lambda_56
tff(fact_10262_ATP_Olambda__57,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hs,Uu2),Uua) = $let(
        m3: real,
        m3:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(plus_plus_real(m3),one_one_real)) ) ).

% ATP.lambda_57
tff(fact_10263_ATP_Olambda__58,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_hr,Uu2),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m3),one_one_rat)) ) ).

% ATP.lambda_58
tff(fact_10264_ATP_Olambda__59,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu,Uu2),Uua) = $let(
        m3: nat,
        m3:= 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(semiri1316708129612266289at_nat,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_59
tff(fact_10265_ATP_Olambda__60,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ht,Uu2),Uua) = $let(
        m3: int,
        m3:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_60
tff(fact_10266_ATP_Olambda__61,axiom,
    ! [Uu2: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_zo,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_fst_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu2)),product_snd_int_int(Uua))) ).

% ATP.lambda_61
tff(fact_10267_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_hx(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(power_power_real(Uu2),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_62
tff(fact_10268_ATP_Olambda__63,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_xk(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(power_power_real(Uu2),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_63
tff(fact_10269_ATP_Olambda__64,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xp(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_64
tff(fact_10270_ATP_Olambda__65,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vk,Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_65
tff(fact_10271_ATP_Olambda__66,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_vi(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_66
tff(fact_10272_ATP_Olambda__67,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vj,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_67
tff(fact_10273_ATP_Olambda__68,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rw(Uu2),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_68
tff(fact_10274_ATP_Olambda__69,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_ry(Uu2),Uua) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_69
tff(fact_10275_ATP_Olambda__70,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rx(Uu2),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_70
tff(fact_10276_ATP_Olambda__71,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pk(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu2),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu2),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_71
tff(fact_10277_ATP_Olambda__72,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_pm(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(Uu2),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_72
tff(fact_10278_ATP_Olambda__73,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pl(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu2),Uua)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uu2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_73
tff(fact_10279_ATP_Olambda__74,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_tg(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu2),Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)) ).

% ATP.lambda_74
tff(fact_10280_ATP_Olambda__75,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_ti(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(Uu2),Uua)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_75
tff(fact_10281_ATP_Olambda__76,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_th(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu2),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ).

% ATP.lambda_76
tff(fact_10282_ATP_Olambda__77,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tt,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_77
tff(fact_10283_ATP_Olambda__78,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_pc(Uu2),Uua) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_78
tff(fact_10284_ATP_Olambda__79,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ye,Uu2),Uua) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uua),Uu2)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_79
tff(fact_10285_ATP_Olambda__80,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zd,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_80
tff(fact_10286_ATP_Olambda__81,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rr,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uu2) ).

% ATP.lambda_81
tff(fact_10287_ATP_Olambda__82,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rq,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_82
tff(fact_10288_ATP_Olambda__83,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ev(Uu2),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).

% ATP.lambda_83
tff(fact_10289_ATP_Olambda__84,axiom,
    ! [Uu2: nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_eu(Uu2),Uua)
    <=> ( aa_nat_real(power_power_real(Uua),Uu2) = one_one_real ) ) ).

% ATP.lambda_84
tff(fact_10290_ATP_Olambda__85,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_xx(Uu2),Uua) = aa_nat_real(power_power_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_85
tff(fact_10291_ATP_Olambda__86,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_hv(Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(power_power_real(Uu2),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_86
tff(fact_10292_ATP_Olambda__87,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qu(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_87
tff(fact_10293_ATP_Olambda__88,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qv(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_88
tff(fact_10294_ATP_Olambda__89,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qw(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_89
tff(fact_10295_ATP_Olambda__90,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qx(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_90
tff(fact_10296_ATP_Olambda__91,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mf(Uu2),Uua) = aa_real_real(plus_plus_real(aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_91
tff(fact_10297_ATP_Olambda__92,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mc(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_92
tff(fact_10298_ATP_Olambda__93,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_me(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_93
tff(fact_10299_ATP_Olambda__94,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_md(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_94
tff(fact_10300_ATP_Olambda__95,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ju(Uu2),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,aa_nat_nat(suc,Uua))),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_95
tff(fact_10301_ATP_Olambda__96,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_js(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_96
tff(fact_10302_ATP_Olambda__97,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jt(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,aa_nat_nat(suc,Uua))),aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_97
tff(fact_10303_ATP_Olambda__98,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_hy(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_98
tff(fact_10304_ATP_Olambda__99,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hz(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu2,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_99
tff(fact_10305_ATP_Olambda__100,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ie(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_100
tff(fact_10306_ATP_Olambda__101,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_iu(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_101
tff(fact_10307_ATP_Olambda__102,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ma(Uu2),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_102
tff(fact_10308_ATP_Olambda__103,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ly(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_103
tff(fact_10309_ATP_Olambda__104,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lz(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_104
tff(fact_10310_ATP_Olambda__105,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lx(Uu2),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_105
tff(fact_10311_ATP_Olambda__106,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lv(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_106
tff(fact_10312_ATP_Olambda__107,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lw(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_107
tff(fact_10313_ATP_Olambda__108,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nl(Uu2),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_108
tff(fact_10314_ATP_Olambda__109,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nj(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_109
tff(fact_10315_ATP_Olambda__110,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nk(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_110
tff(fact_10316_ATP_Olambda__111,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qf(Uu2),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu2,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_111
tff(fact_10317_ATP_Olambda__112,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qi(Uu2),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu2,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_112
tff(fact_10318_ATP_Olambda__113,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rn(Uu2),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu2,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_113
tff(fact_10319_ATP_Olambda__114,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_rl(Uu2),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu2,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_114
tff(fact_10320_ATP_Olambda__115,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yd(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_xp(Uu2),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_115
tff(fact_10321_ATP_Olambda__116,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yc(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_xp(Uu2),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_116
tff(fact_10322_ATP_Olambda__117,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ze,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uu2) ).

% ATP.lambda_117
tff(fact_10323_ATP_Olambda__118,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mu(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua)) ).

% ATP.lambda_118
tff(fact_10324_ATP_Olambda__119,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_mt(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu2),Uua)) ).

% ATP.lambda_119
tff(fact_10325_ATP_Olambda__120,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_wp(Uu2),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu2))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_120
tff(fact_10326_ATP_Olambda__121,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xa,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu2)) ).

% ATP.lambda_121
tff(fact_10327_ATP_Olambda__122,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_pe(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu2),Uua)) ).

% ATP.lambda_122
tff(fact_10328_ATP_Olambda__123,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_ou(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu2),Uua)) ).

% ATP.lambda_123
tff(fact_10329_ATP_Olambda__124,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_zm,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu2)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_124
tff(fact_10330_ATP_Olambda__125,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_vw,Uu2),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu2),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_125
tff(fact_10331_ATP_Olambda__126,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_vy,Uu2),Uua) = $let(
        l2: nat,
        l2:= aa_Code_integer_nat(code_nat_of_integer,Uu2),
        $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_126
tff(fact_10332_ATP_Olambda__127,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pn(Uu2),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu2) ).

% ATP.lambda_127
tff(fact_10333_ATP_Olambda__128,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pp,Uu2),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu2) ).

% ATP.lambda_128
tff(fact_10334_ATP_Olambda__129,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_po(Uu2),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu2) ).

% ATP.lambda_129
tff(fact_10335_ATP_Olambda__130,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gw,Uu2),Uua) = produc1086072967326762835nteger(Uu2,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_130
tff(fact_10336_ATP_Olambda__131,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gv,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),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_131
tff(fact_10337_ATP_Olambda__132,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gu,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),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)) ).

% ATP.lambda_132
tff(fact_10338_ATP_Olambda__133,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gz,Uu2),Uua) = produc1086072967326762835nteger(Uu2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_133
tff(fact_10339_ATP_Olambda__134,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gy,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),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_134
tff(fact_10340_ATP_Olambda__135,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gx,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_135
tff(fact_10341_ATP_Olambda__136,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pq(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_136
tff(fact_10342_ATP_Olambda__137,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_ps(Uu2),Uua) = aa_real_real(plus_plus_real(Uu2),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_137
tff(fact_10343_ATP_Olambda__138,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pr(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_138
tff(fact_10344_ATP_Olambda__139,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vh,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)) ).

% ATP.lambda_139
tff(fact_10345_ATP_Olambda__140,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_xy(Uu2),Uua) = aa_real_real(times_times_real(Uu2),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_140
tff(fact_10346_ATP_Olambda__141,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_zv(Uu2),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,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% ATP.lambda_141
tff(fact_10347_ATP_Olambda__142,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_xw(Uu2),Uua) = aa_real_real(plus_plus_real(Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_142
tff(fact_10348_ATP_Olambda__143,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_xt(Uu2),Uua) = aa_real_real(plus_plus_real(Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_143
tff(fact_10349_ATP_Olambda__144,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_tu(Uu2),Uua) = aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_144
tff(fact_10350_ATP_Olambda__145,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_tw(Uu2),Uua) = aa_real_real(minus_minus_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_145
tff(fact_10351_ATP_Olambda__146,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_tv(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_146
tff(fact_10352_ATP_Olambda__147,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tx,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_147
tff(fact_10353_ATP_Olambda__148,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_ty(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_148
tff(fact_10354_ATP_Olambda__149,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_sj(Uu2),Uua) = aa_real_real(plus_plus_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_149
tff(fact_10355_ATP_Olambda__150,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_si(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_150
tff(fact_10356_ATP_Olambda__151,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sk,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_151
tff(fact_10357_ATP_Olambda__152,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_sl(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_152
tff(fact_10358_ATP_Olambda__153,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_zt,Uu2),Uua) = cons_nat(Uu2,aa_nat_list_nat(nat_list_decode,Uua)) ).

% ATP.lambda_153
tff(fact_10359_ATP_Olambda__154,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wo,Uu2),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu2) ) ) ).

% ATP.lambda_154
tff(fact_10360_ATP_Olambda__155,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_es,Uu2),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uua),Uu2) ) ).

% ATP.lambda_155
tff(fact_10361_ATP_Olambda__156,axiom,
    ! [Uu2: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_er(Uu2),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu2) ) ).

% ATP.lambda_156
tff(fact_10362_ATP_Olambda__157,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_et(Uu2),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu2) ) ).

% ATP.lambda_157
tff(fact_10363_ATP_Olambda__158,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_ql,Uu2),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu2) ) ).

% ATP.lambda_158
tff(fact_10364_ATP_Olambda__159,axiom,
    ! [Uu2: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_qm,Uu2),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu2) ) ).

% ATP.lambda_159
tff(fact_10365_ATP_Olambda__160,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_eq,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu2) ) ).

% ATP.lambda_160
tff(fact_10366_ATP_Olambda__161,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qn,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu2) ) ).

% ATP.lambda_161
tff(fact_10367_ATP_Olambda__162,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_mb,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2) ) ).

% ATP.lambda_162
tff(fact_10368_ATP_Olambda__163,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wq,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu2) ).

% ATP.lambda_163
tff(fact_10369_ATP_Olambda__164,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xb,Uu2),Uua) = aa_nat_real(power_power_real(Uua),Uu2) ).

% ATP.lambda_164
tff(fact_10370_ATP_Olambda__165,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yy,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ).

% ATP.lambda_165
tff(fact_10371_ATP_Olambda__166,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ft,Uu2),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu2) ) ).

% ATP.lambda_166
tff(fact_10372_ATP_Olambda__167,axiom,
    ! [Uu2: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_fr,Uu2),Uua)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Uua),Uu2) ) ).

% ATP.lambda_167
tff(fact_10373_ATP_Olambda__168,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fu,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu2) ) ).

% ATP.lambda_168
tff(fact_10374_ATP_Olambda__169,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fs,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu2) ) ).

% ATP.lambda_169
tff(fact_10375_ATP_Olambda__170,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_wf,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu2) ).

% ATP.lambda_170
tff(fact_10376_ATP_Olambda__171,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rk,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu2) ).

% ATP.lambda_171
tff(fact_10377_ATP_Olambda__172,axiom,
    ! [Uu2: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_be(Uu2),Uua)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uua),Uu2) ) ).

% ATP.lambda_172
tff(fact_10378_ATP_Olambda__173,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ae(Uu2),Uua)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu2) ) ).

% ATP.lambda_173
tff(fact_10379_ATP_Olambda__174,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ad(Uu2),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu2) ) ).

% ATP.lambda_174
tff(fact_10380_ATP_Olambda__175,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_a(Uu2),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu2) ) ).

% ATP.lambda_175
tff(fact_10381_ATP_Olambda__176,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ag(Uu2),Uua)
    <=> member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_176
tff(fact_10382_ATP_Olambda__177,axiom,
    ! [Uu2: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_aa(Uu2),(Uua))
    <=> aa_set_o_o(member_o((Uua)),Uu2) ) ).

% ATP.lambda_177
tff(fact_10383_ATP_Olambda__178,axiom,
    ! [Uu2: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_af(Uu2),Uua)
    <=> member_set_nat(Uua,Uu2) ) ).

% ATP.lambda_178
tff(fact_10384_ATP_Olambda__179,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ac(Uu2),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu2) ) ).

% ATP.lambda_179
tff(fact_10385_ATP_Olambda__180,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ab(Uu2),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu2) ) ).

% ATP.lambda_180
tff(fact_10386_ATP_Olambda__181,axiom,
    ! [Uu2: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_aj(Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_181
tff(fact_10387_ATP_Olambda__182,axiom,
    ! [Uu2: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_an,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_182
tff(fact_10388_ATP_Olambda__183,axiom,
    ! [Uu2: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ak,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_183
tff(fact_10389_ATP_Olambda__184,axiom,
    ! [Uu2: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_am(Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_184
tff(fact_10390_ATP_Olambda__185,axiom,
    ! [Uu2: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_ao,(Uu2)),(Uua))
    <=> ( (Uua)
      <=> (Uu2) ) ) ).

% ATP.lambda_185
tff(fact_10391_ATP_Olambda__186,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_al(Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_186
tff(fact_10392_ATP_Olambda__187,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ap,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_187
tff(fact_10393_ATP_Olambda__188,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aq,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_188
tff(fact_10394_ATP_Olambda__189,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zi(Uu2),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu2) ).

% ATP.lambda_189
tff(fact_10395_ATP_Olambda__190,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ol(Uu2),Uua) = aa_nat_real(Uu2,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_190
tff(fact_10396_ATP_Olambda__191,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ok(Uu2),Uua) = aa_nat_real(Uu2,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_191
tff(fact_10397_ATP_Olambda__192,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_il(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_192
tff(fact_10398_ATP_Olambda__193,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ys(Uu2),Uua)
    <=> aa_nat_o(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_193
tff(fact_10399_ATP_Olambda__194,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jq(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_194
tff(fact_10400_ATP_Olambda__195,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_is(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_195
tff(fact_10401_ATP_Olambda__196,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jr(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_196
tff(fact_10402_ATP_Olambda__197,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_wx(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_197
tff(fact_10403_ATP_Olambda__198,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_wv(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_198
tff(fact_10404_ATP_Olambda__199,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_zr,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_zq(Uu2),Uua)) ).

% ATP.lambda_199
tff(fact_10405_ATP_Olambda__200,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_wn,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wm(Uu2),Uua)) ).

% ATP.lambda_200
tff(fact_10406_ATP_Olambda__201,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_wl,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wk(Uu2),Uua)) ).

% ATP.lambda_201
tff(fact_10407_ATP_Olambda__202,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_wj,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wi(Uu2),Uua)) ).

% ATP.lambda_202
tff(fact_10408_ATP_Olambda__203,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_wh,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wg(Uu2),Uua)) ).

% ATP.lambda_203
tff(fact_10409_ATP_Olambda__204,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_we,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wd(Uu2),Uua)) ).

% ATP.lambda_204
tff(fact_10410_ATP_Olambda__205,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_xn(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xm(Uu2),Uua)) ).

% ATP.lambda_205
tff(fact_10411_ATP_Olambda__206,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_vo(Uu2),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).

% ATP.lambda_206
tff(fact_10412_ATP_Olambda__207,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fv,Uu2),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_207
tff(fact_10413_ATP_Olambda__208,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pt(Uu2),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_208
tff(fact_10414_ATP_Olambda__209,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_pv(Uu2),Uua) = times_times_real(aa_real_real(plus_plus_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_209
tff(fact_10415_ATP_Olambda__210,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_pu(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_210
tff(fact_10416_ATP_Olambda__211,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_px(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_211
tff(fact_10417_ATP_Olambda__212,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_pw(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_212
tff(fact_10418_ATP_Olambda__213,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zg(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).

% ATP.lambda_213
tff(fact_10419_ATP_Olambda__214,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aTP_Lamm_xv(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(Uu2),Uua)) ).

% ATP.lambda_214
tff(fact_10420_ATP_Olambda__215,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_zw(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)) ).

% ATP.lambda_215
tff(fact_10421_ATP_Olambda__216,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu2),Uua)) ).

% ATP.lambda_216
tff(fact_10422_ATP_Olambda__217,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yx,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu2)) ).

% ATP.lambda_217
tff(fact_10423_ATP_Olambda__218,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vu,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu2),Uua)) ).

% ATP.lambda_218
tff(fact_10424_ATP_Olambda__219,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vv,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu2)) ).

% ATP.lambda_219
tff(fact_10425_ATP_Olambda__220,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_hk(Uu2),Uua)
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu2) ) ).

% ATP.lambda_220
tff(fact_10426_ATP_Olambda__221,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_hj(Uu2),Uua)
    <=> ~ aa_set_complex_o(member_complex(Uua),Uu2) ) ).

% ATP.lambda_221
tff(fact_10427_ATP_Olambda__222,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_hf(Uu2),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu2) ) ).

% ATP.lambda_222
tff(fact_10428_ATP_Olambda__223,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_hm(Uu2),Uua)
    <=> ~ member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_223
tff(fact_10429_ATP_Olambda__224,axiom,
    ! [Uu2: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_hg(Uu2),(Uua))
    <=> ~ aa_set_o_o(member_o((Uua)),Uu2) ) ).

% ATP.lambda_224
tff(fact_10430_ATP_Olambda__225,axiom,
    ! [Uu2: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hl(Uu2),Uua)
    <=> ~ member_set_nat(Uua,Uu2) ) ).

% ATP.lambda_225
tff(fact_10431_ATP_Olambda__226,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_hi(Uu2),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu2) ) ).

% ATP.lambda_226
tff(fact_10432_ATP_Olambda__227,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_hh(Uu2),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu2) ) ).

% ATP.lambda_227
tff(fact_10433_ATP_Olambda__228,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_iz(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu2,Uua)) ).

% ATP.lambda_228
tff(fact_10434_ATP_Olambda__229,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xr(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_229
tff(fact_10435_ATP_Olambda__230,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_sx(Uu2),Uua) = plus_plus_real(aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_230
tff(fact_10436_ATP_Olambda__231,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_su(Uu2),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_231
tff(fact_10437_ATP_Olambda__232,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_sw(Uu2),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_232
tff(fact_10438_ATP_Olambda__233,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_sv(Uu2),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_233
tff(fact_10439_ATP_Olambda__234,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ic(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_234
tff(fact_10440_ATP_Olambda__235,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_id(Uu2),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_235
tff(fact_10441_ATP_Olambda__236,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yb(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_236
tff(fact_10442_ATP_Olambda__237,axiom,
    ! [Uu2: fun_Pr4796943543331071514_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_hc(Uu2),Uua)
    <=> ~ aa_Pro6150637604445964947_nat_o(Uu2,Uua) ) ).

% ATP.lambda_237
tff(fact_10443_ATP_Olambda__238,axiom,
    ! [Uu2: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_hb(Uu2),Uua)
    <=> ~ aa_complex_o(Uu2,Uua) ) ).

% ATP.lambda_238
tff(fact_10444_ATP_Olambda__239,axiom,
    ! [Uu2: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_he(Uu2),Uua)
    <=> ~ aa_list_nat_o(Uu2,Uua) ) ).

% ATP.lambda_239
tff(fact_10445_ATP_Olambda__240,axiom,
    ! [Uu2: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hd(Uu2),Uua)
    <=> ~ aa_set_nat_o(Uu2,Uua) ) ).

% ATP.lambda_240
tff(fact_10446_ATP_Olambda__241,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ha(Uu2),Uua)
    <=> ~ aa_nat_o(Uu2,Uua) ) ).

% ATP.lambda_241
tff(fact_10447_ATP_Olambda__242,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zh(Uu2),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mb,Uu2)) ).

% ATP.lambda_242
tff(fact_10448_ATP_Olambda__243,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oi(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_243
tff(fact_10449_ATP_Olambda__244,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oj(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu2,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_244
tff(fact_10450_ATP_Olambda__245,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fy(Uu2),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu2)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,aa_num_Code_integer(numera6620942414471956472nteger,Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_245
tff(fact_10451_ATP_Olambda__246,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_fw(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu2)),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,Uu2))),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_246
tff(fact_10452_ATP_Olambda__247,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fx(Uu2),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu2))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_247
tff(fact_10453_ATP_Olambda__248,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_wb(Uu2),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,Uu2),Uub))) ).

% ATP.lambda_248
tff(fact_10454_ATP_Olambda__249,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_wc(Uu2),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(abs_abs_Code_integer(Uu2),Uub))) ).

% ATP.lambda_249
tff(fact_10455_ATP_Olambda__250,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_wa(Uu2),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(Uu2,Uub))) ).

% ATP.lambda_250
tff(fact_10456_ATP_Olambda__251,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sc(Uu2),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_251
tff(fact_10457_ATP_Olambda__252,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sa(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_252
tff(fact_10458_ATP_Olambda__253,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ro(Uu2),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_253
tff(fact_10459_ATP_Olambda__254,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qg(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu2,Uub),Uua) ).

% ATP.lambda_254
tff(fact_10460_ATP_Olambda__255,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qj(Uu2),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu2,Uub),Uua) ).

% ATP.lambda_255
tff(fact_10461_ATP_Olambda__256,axiom,
    ! [Uu2: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ne(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu2,one_one_complex,zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_256
tff(fact_10462_ATP_Olambda__257,axiom,
    ! [Uu2: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nf(Uu2),Uua),Uub) = aa_real_real(times_times_real($ite(Uub = Uu2,one_one_real,zero_zero_real)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_257
tff(fact_10463_ATP_Olambda__258,axiom,
    ! [Uu2: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ng(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu2,one_one_int,zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_258
tff(fact_10464_ATP_Olambda__259,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_vx(Uu2),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu2),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_259
tff(fact_10465_ATP_Olambda__260,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qk(Uu2),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qj(Uu2),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_260
tff(fact_10466_ATP_Olambda__261,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qh(Uu2),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qg(Uu2),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_261
tff(fact_10467_ATP_Olambda__262,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rp(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ro(Uu2),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_262
tff(fact_10468_ATP_Olambda__263,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rm(Uu2),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qg(Uu2),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_263
tff(fact_10469_ATP_Olambda__264,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_yo(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yn(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_264
tff(fact_10470_ATP_Olambda__265,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ym(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yl(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_265
tff(fact_10471_ATP_Olambda__266,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_yk(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yj(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_266
tff(fact_10472_ATP_Olambda__267,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_yi(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yh(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_267
tff(fact_10473_ATP_Olambda__268,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xd(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_268
tff(fact_10474_ATP_Olambda__269,axiom,
    ! [Uu2: real,Uua: fun_na2180513548729900188l_real,Uub: nat] : aa_nat_real(aa_fun4674562053777771865t_real(aTP_Lamm_ow(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_literal_real(aa_nat8303615512603269253l_real(Uua,Uub),zero_zero_literal)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_269
tff(fact_10475_ATP_Olambda__270,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ox(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_270
tff(fact_10476_ATP_Olambda__271,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_oy(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_271
tff(fact_10477_ATP_Olambda__272,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_oz(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_272
tff(fact_10478_ATP_Olambda__273,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_pa(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_273
tff(fact_10479_ATP_Olambda__274,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_sg(Uu2),Uua),Uub)
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_iq(Uu2),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_274
tff(fact_10480_ATP_Olambda__275,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_sh(Uu2),Uua),Uub)
    <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_io(Uu2),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_275
tff(fact_10481_ATP_Olambda__276,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_if(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_276
tff(fact_10482_ATP_Olambda__277,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ih(Uu2),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(power_power_real(zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_277
tff(fact_10483_ATP_Olambda__278,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ig(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_278
tff(fact_10484_ATP_Olambda__279,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xl(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_279
tff(fact_10485_ATP_Olambda__280,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pb(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_280
tff(fact_10486_ATP_Olambda__281,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_zb(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_281
tff(fact_10487_ATP_Olambda__282,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_op(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu2,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_282
tff(fact_10488_ATP_Olambda__283,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_or(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu2,Uub))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_283
tff(fact_10489_ATP_Olambda__284,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_zc(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_284
tff(fact_10490_ATP_Olambda__285,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_em(Uu2),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_285
tff(fact_10491_ATP_Olambda__286,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_el(Uu2),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua) ) ) ).

% ATP.lambda_286
tff(fact_10492_ATP_Olambda__287,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_ek(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_287
tff(fact_10493_ATP_Olambda__288,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_en(Uu2),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(set_o2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_288
tff(fact_10494_ATP_Olambda__289,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ep(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_289
tff(fact_10495_ATP_Olambda__290,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_eo(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_290
tff(fact_10496_ATP_Olambda__291,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_eg(Uu2),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu2)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_291
tff(fact_10497_ATP_Olambda__292,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_ef(Uu2),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Uub)),Uu2)
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_292
tff(fact_10498_ATP_Olambda__293,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_ee(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_293
tff(fact_10499_ATP_Olambda__294,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_eh(Uu2),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(set_o2(Uub)),Uu2)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_294
tff(fact_10500_ATP_Olambda__295,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ej(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu2)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_295
tff(fact_10501_ATP_Olambda__296,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_ei(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu2)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_296
tff(fact_10502_ATP_Olambda__297,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_za(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_297
tff(fact_10503_ATP_Olambda__298,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vs(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_298
tff(fact_10504_ATP_Olambda__299,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zf(Uu2),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_299
tff(fact_10505_ATP_Olambda__300,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vt(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_300
tff(fact_10506_ATP_Olambda__301,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_oo(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu2),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_301
tff(fact_10507_ATP_Olambda__302,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oq(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uu2),Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_302
tff(fact_10508_ATP_Olambda__303,axiom,
    ! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ot(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu2),Uub)) ).

% ATP.lambda_303
tff(fact_10509_ATP_Olambda__304,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_os(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_304
tff(fact_10510_ATP_Olambda__305,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_se(Uu2),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,Uu2),Uub)) ).

% ATP.lambda_305
tff(fact_10511_ATP_Olambda__306,axiom,
    ! [Uu2: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_bn(Uu2),Uua),Uub)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uu2)
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_306
tff(fact_10512_ATP_Olambda__307,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_bj(Uu2),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu2)
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_307
tff(fact_10513_ATP_Olambda__308,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_bi(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_308
tff(fact_10514_ATP_Olambda__309,axiom,
    ! [Uu2: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_bf(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_309
tff(fact_10515_ATP_Olambda__310,axiom,
    ! [Uu2: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_bl(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_310
tff(fact_10516_ATP_Olambda__311,axiom,
    ! [Uu2: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_bg(Uu2),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu2)
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_311
tff(fact_10517_ATP_Olambda__312,axiom,
    ! [Uu2: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_bk(Uu2),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu2)
        | member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_312
tff(fact_10518_ATP_Olambda__313,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bm(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_313
tff(fact_10519_ATP_Olambda__314,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bh(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_314
tff(fact_10520_ATP_Olambda__315,axiom,
    ! [Uu2: produc3843707927480180839at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_cr(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_315
tff(fact_10521_ATP_Olambda__316,axiom,
    ! [Uu2: product_prod_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_cx(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_316
tff(fact_10522_ATP_Olambda__317,axiom,
    ! [Uu2: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_cw(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_317
tff(fact_10523_ATP_Olambda__318,axiom,
    ! [Uu2: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_cs(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_318
tff(fact_10524_ATP_Olambda__319,axiom,
    ! [Uu2: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_cz(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_319
tff(fact_10525_ATP_Olambda__320,axiom,
    ! [Uu2: $o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_ct((Uu2)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu2) )
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_320
tff(fact_10526_ATP_Olambda__321,axiom,
    ! [Uu2: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_cy(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_321
tff(fact_10527_ATP_Olambda__322,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_cv(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_322
tff(fact_10528_ATP_Olambda__323,axiom,
    ! [Uu2: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_cu(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_323
tff(fact_10529_ATP_Olambda__324,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ah(Uu2),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu2),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_324
tff(fact_10530_ATP_Olambda__325,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ai(Uu2),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu2),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_325
tff(fact_10531_ATP_Olambda__326,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_ec(Uu2),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu2)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_326
tff(fact_10532_ATP_Olambda__327,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_dy(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_327
tff(fact_10533_ATP_Olambda__328,axiom,
    ! [Uu2: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_dv(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_328
tff(fact_10534_ATP_Olambda__329,axiom,
    ! [Uu2: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ea(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_329
tff(fact_10535_ATP_Olambda__330,axiom,
    ! [Uu2: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_dw(Uu2),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu2)
        & aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_330
tff(fact_10536_ATP_Olambda__331,axiom,
    ! [Uu2: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_dz(Uu2),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu2)
        & member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_331
tff(fact_10537_ATP_Olambda__332,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_eb(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_332
tff(fact_10538_ATP_Olambda__333,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_dx(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_333
tff(fact_10539_ATP_Olambda__334,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_by(Uu2),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu2)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_334
tff(fact_10540_ATP_Olambda__335,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bx(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_335
tff(fact_10541_ATP_Olambda__336,axiom,
    ! [Uu2: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bu(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_336
tff(fact_10542_ATP_Olambda__337,axiom,
    ! [Uu2: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ca(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_337
tff(fact_10543_ATP_Olambda__338,axiom,
    ! [Uu2: set_o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bv(Uu2),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu2)
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_338
tff(fact_10544_ATP_Olambda__339,axiom,
    ! [Uu2: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bz(Uu2),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu2)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_339
tff(fact_10545_ATP_Olambda__340,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_340
tff(fact_10546_ATP_Olambda__341,axiom,
    ! [Uu2: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bw(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_341
tff(fact_10547_ATP_Olambda__342,axiom,
    ! [Uu2: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_da(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_342
tff(fact_10548_ATP_Olambda__343,axiom,
    ! [Uu2: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_de(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_343
tff(fact_10549_ATP_Olambda__344,axiom,
    ! [Uu2: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_db(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_344
tff(fact_10550_ATP_Olambda__345,axiom,
    ! [Uu2: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_dd(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_345
tff(fact_10551_ATP_Olambda__346,axiom,
    ! [Uu2: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_df((Uu2)),Uua),(Uub))
    <=> ( ( (Uu2)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_346
tff(fact_10552_ATP_Olambda__347,axiom,
    ! [Uu2: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dc(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_347
tff(fact_10553_ATP_Olambda__348,axiom,
    ! [Uu2: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dg(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_348
tff(fact_10554_ATP_Olambda__349,axiom,
    ! [Uu2: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dh(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_349
tff(fact_10555_ATP_Olambda__350,axiom,
    ! [Uu2: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_di(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_350
tff(fact_10556_ATP_Olambda__351,axiom,
    ! [Uu2: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_dm(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_351
tff(fact_10557_ATP_Olambda__352,axiom,
    ! [Uu2: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_dj(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_352
tff(fact_10558_ATP_Olambda__353,axiom,
    ! [Uu2: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_dl(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_353
tff(fact_10559_ATP_Olambda__354,axiom,
    ! [Uu2: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_dn((Uu2)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu2) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_354
tff(fact_10560_ATP_Olambda__355,axiom,
    ! [Uu2: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dk(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_355
tff(fact_10561_ATP_Olambda__356,axiom,
    ! [Uu2: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_do(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_356
tff(fact_10562_ATP_Olambda__357,axiom,
    ! [Uu2: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dp(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_357
tff(fact_10563_ATP_Olambda__358,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ff(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_358
tff(fact_10564_ATP_Olambda__359,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aTP_Lamm_fp(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & ( aa_complex_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_359
tff(fact_10565_ATP_Olambda__360,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_ex(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_360
tff(fact_10566_ATP_Olambda__361,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_fh(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_361
tff(fact_10567_ATP_Olambda__362,axiom,
    ! [Uu2: set_o,Uua: fun_o_real,Uub: $o] :
      ( aa_o_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_ez(Uu2),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu2)
        & ( aa_o_real(Uua,(Uub)) != zero_zero_real ) ) ) ).

% ATP.lambda_362
tff(fact_10568_ATP_Olambda__363,axiom,
    ! [Uu2: set_o,Uua: fun_o_rat,Uub: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_fj(Uu2),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu2)
        & ( aa_o_rat(Uua,(Uub)) != zero_zero_rat ) ) ) ).

% ATP.lambda_363
tff(fact_10569_ATP_Olambda__364,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_fd(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_364
tff(fact_10570_ATP_Olambda__365,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_fn(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_365
tff(fact_10571_ATP_Olambda__366,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_fb(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_366
tff(fact_10572_ATP_Olambda__367,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_fl(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_367
tff(fact_10573_ATP_Olambda__368,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_av(Uu2),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu2)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_368
tff(fact_10574_ATP_Olambda__369,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_au(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_369
tff(fact_10575_ATP_Olambda__370,axiom,
    ! [Uu2: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_ar(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_370
tff(fact_10576_ATP_Olambda__371,axiom,
    ! [Uu2: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ax(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_371
tff(fact_10577_ATP_Olambda__372,axiom,
    ! [Uu2: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_as(Uu2),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu2)
        & ~ aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_372
tff(fact_10578_ATP_Olambda__373,axiom,
    ! [Uu2: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_aw(Uu2),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu2)
        & ~ member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_373
tff(fact_10579_ATP_Olambda__374,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ay(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_374
tff(fact_10580_ATP_Olambda__375,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_at(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_375
tff(fact_10581_ATP_Olambda__376,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yu(Uu2),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)),Uu2) ) ).

% ATP.lambda_376
tff(fact_10582_ATP_Olambda__377,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rz(Uu2),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)),Uu2) ) ).

% ATP.lambda_377
tff(fact_10583_ATP_Olambda__378,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ss(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu2) ) ).

% ATP.lambda_378
tff(fact_10584_ATP_Olambda__379,axiom,
    ! [Uu2: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_bd(Uu2),Uua),Uub)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub)),Uu2) ) ).

% ATP.lambda_379
tff(fact_10585_ATP_Olambda__380,axiom,
    ! [Uu2: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_az(Uu2),Uua),Uub)
    <=> member7279096912039735102um_num(product_Pair_num_num(Uua,Uub),Uu2) ) ).

% ATP.lambda_380
tff(fact_10586_ATP_Olambda__381,axiom,
    ! [Uu2: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_ba(Uu2),Uua),Uub)
    <=> member9148766508732265716at_num(product_Pair_nat_num(Uua,Uub),Uu2) ) ).

% ATP.lambda_381
tff(fact_10587_ATP_Olambda__382,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bb(Uu2),Uua),Uub)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu2) ) ).

% ATP.lambda_382
tff(fact_10588_ATP_Olambda__383,axiom,
    ! [Uu2: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bc(Uu2),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu2) ) ).

% ATP.lambda_383
tff(fact_10589_ATP_Olambda__384,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ew(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).

% ATP.lambda_384
tff(fact_10590_ATP_Olambda__385,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_vp(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_385
tff(fact_10591_ATP_Olambda__386,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_386
tff(fact_10592_ATP_Olambda__387,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iw(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu2,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_387
tff(fact_10593_ATP_Olambda__388,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_ya(Uu2),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_388
tff(fact_10594_ATP_Olambda__389,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xm(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_389
tff(fact_10595_ATP_Olambda__390,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_iq(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_390
tff(fact_10596_ATP_Olambda__391,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_io(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_391
tff(fact_10597_ATP_Olambda__392,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_sm(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_392
tff(fact_10598_ATP_Olambda__393,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tq(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_393
tff(fact_10599_ATP_Olambda__394,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_sn(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_394
tff(fact_10600_ATP_Olambda__395,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vq(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10601_ATP_Olambda__396,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_lk(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu2,Uub)) ).

% ATP.lambda_396
tff(fact_10602_ATP_Olambda__397,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_lp(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(Uua,Uub)),aa_complex_real(Uu2,Uub)) ).

% ATP.lambda_397
tff(fact_10603_ATP_Olambda__398,axiom,
    ! [Uu2: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_lt(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu2,Uub)) ).

% ATP.lambda_398
tff(fact_10604_ATP_Olambda__399,axiom,
    ! [Uu2: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_ll(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu2,Uub)) ).

% ATP.lambda_399
tff(fact_10605_ATP_Olambda__400,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_lq(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(Uua,Uub)),aa_real_real(Uu2,Uub)) ).

% ATP.lambda_400
tff(fact_10606_ATP_Olambda__401,axiom,
    ! [Uu2: fun_o_Code_integer,Uua: fun_o_Code_integer,Uub: $o] : aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_lm(Uu2),Uua),(Uub)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(Uua,(Uub))),aa_o_Code_integer(Uu2,(Uub))) ).

% ATP.lambda_401
tff(fact_10607_ATP_Olambda__402,axiom,
    ! [Uu2: fun_o_real,Uua: fun_o_real,Uub: $o] : aa_o_real(aa_fun688608109223240316o_real(aTP_Lamm_lr(Uu2),Uua),(Uub)) = aa_real_real(times_times_real(aa_o_real(Uua,(Uub))),aa_o_real(Uu2,(Uub))) ).

% ATP.lambda_402
tff(fact_10608_ATP_Olambda__403,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ln(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu2,Uub)) ).

% ATP.lambda_403
tff(fact_10609_ATP_Olambda__404,axiom,
    ! [Uu2: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_lo(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu2,Uub)) ).

% ATP.lambda_404
tff(fact_10610_ATP_Olambda__405,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ls(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_int_real(Uua,Uub)),aa_int_real(Uu2,Uub)) ).

% ATP.lambda_405
tff(fact_10611_ATP_Olambda__406,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xq(Uu2),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_406
tff(fact_10612_ATP_Olambda__407,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_mx(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu2,Uub)) ).

% ATP.lambda_407
tff(fact_10613_ATP_Olambda__408,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_mv(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu2,Uub)) ).

% ATP.lambda_408
tff(fact_10614_ATP_Olambda__409,axiom,
    ! [Uu2: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_my(Uu2),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_o_nat(Uua,(Uub))),aa_o_nat(Uu2,(Uub))) ).

% ATP.lambda_409
tff(fact_10615_ATP_Olambda__410,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nh(Uu2),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uua,Uub)),aa_nat_real(Uu2,Uub)) ).

% ATP.lambda_410
tff(fact_10616_ATP_Olambda__411,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mw(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu2,Uub)) ).

% ATP.lambda_411
tff(fact_10617_ATP_Olambda__412,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_mz(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu2,Uub)) ).

% ATP.lambda_412
tff(fact_10618_ATP_Olambda__413,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_im(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_413
tff(fact_10619_ATP_Olambda__414,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(Uu2),Uua),Uub) = aa_real_real(plus_plus_real(aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_414
tff(fact_10620_ATP_Olambda__415,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_415
tff(fact_10621_ATP_Olambda__416,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ik(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_416
tff(fact_10622_ATP_Olambda__417,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_in(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_417
tff(fact_10623_ATP_Olambda__418,axiom,
    ! [Uu2: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_gt(Uu2),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu2,Uub)
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_418
tff(fact_10624_ATP_Olambda__419,axiom,
    ! [Uu2: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_gp(Uu2),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu2,Uub)
       => aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_419
tff(fact_10625_ATP_Olambda__420,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_go(Uu2),Uua),Uub)
    <=> ( aa_complex_o(Uu2,Uub)
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_420
tff(fact_10626_ATP_Olambda__421,axiom,
    ! [Uu2: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_gr(Uu2),Uua),Uub)
    <=> ( aa_list_nat_o(Uu2,Uub)
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_421
tff(fact_10627_ATP_Olambda__422,axiom,
    ! [Uu2: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_gq(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(Uu2,Uub)
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_422
tff(fact_10628_ATP_Olambda__423,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_gs(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_423
tff(fact_10629_ATP_Olambda__424,axiom,
    ! [Uu2: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_bt(Uu2),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu2,Uub)
        | aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_424
tff(fact_10630_ATP_Olambda__425,axiom,
    ! [Uu2: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_bp(Uu2),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu2,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_425
tff(fact_10631_ATP_Olambda__426,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bo(Uu2),Uua),Uub)
    <=> ( aa_complex_o(Uu2,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_426
tff(fact_10632_ATP_Olambda__427,axiom,
    ! [Uu2: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_br(Uu2),Uua),Uub)
    <=> ( aa_list_nat_o(Uu2,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_427
tff(fact_10633_ATP_Olambda__428,axiom,
    ! [Uu2: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bq(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(Uu2,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_428
tff(fact_10634_ATP_Olambda__429,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bs(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_429
tff(fact_10635_ATP_Olambda__430,axiom,
    ! [Uu2: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_du(Uu2),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu2,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_430
tff(fact_10636_ATP_Olambda__431,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_dq(Uu2),Uua),Uub)
    <=> ( aa_complex_o(Uu2,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_431
tff(fact_10637_ATP_Olambda__432,axiom,
    ! [Uu2: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ds(Uu2),Uua),Uub)
    <=> ( aa_list_nat_o(Uu2,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_432
tff(fact_10638_ATP_Olambda__433,axiom,
    ! [Uu2: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dr(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(Uu2,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_433
tff(fact_10639_ATP_Olambda__434,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dt(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_434
tff(fact_10640_ATP_Olambda__435,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ed(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_435
tff(fact_10641_ATP_Olambda__436,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: code_integer,Uub: complex] : aa_com4217951221296830281nteger(aa_Cod1660996421839551523nteger(aTP_Lamm_jv(Uu2),Uua),Uub) = divide6298287555418463151nteger(aa_com4217951221296830281nteger(Uu2,Uub),Uua) ).

% ATP.lambda_436
tff(fact_10642_ATP_Olambda__437,axiom,
    ! [Uu2: fun_complex_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_kb(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_complex_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_437
tff(fact_10643_ATP_Olambda__438,axiom,
    ! [Uu2: fun_complex_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ke(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_complex_int(Uu2,Uub)),Uua) ).

% ATP.lambda_438
tff(fact_10644_ATP_Olambda__439,axiom,
    ! [Uu2: fun_li3231056112175589690nteger,Uua: code_integer,Uub: list_nat] : aa_lis9097500579518594299nteger(aa_Cod1658353545975407189nteger(aTP_Lamm_kt(Uu2),Uua),Uub) = divide6298287555418463151nteger(aa_lis9097500579518594299nteger(Uu2,Uub),Uua) ).

% ATP.lambda_439
tff(fact_10645_ATP_Olambda__440,axiom,
    ! [Uu2: fun_se7586849273513198688nteger,Uua: code_integer,Uub: set_nat] : aa_set871585252387937313nteger(aa_Cod3249490220333259515nteger(aTP_Lamm_kq(Uu2),Uua),Uub) = divide6298287555418463151nteger(aa_set871585252387937313nteger(Uu2,Uub),Uua) ).

% ATP.lambda_440
tff(fact_10646_ATP_Olambda__441,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: nat,Uub: set_nat] : aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_kw(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_441
tff(fact_10647_ATP_Olambda__442,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_jy(Uu2),Uua),Uub) = divide6298287555418463151nteger(aa_nat_Code_integer(Uu2,Uub),Uua) ).

% ATP.lambda_442
tff(fact_10648_ATP_Olambda__443,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kk(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_443
tff(fact_10649_ATP_Olambda__444,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_kh(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_444
tff(fact_10650_ATP_Olambda__445,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_kn(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_445
tff(fact_10651_ATP_Olambda__446,axiom,
    ! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xc(Uu2),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_446
tff(fact_10652_ATP_Olambda__447,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qc(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_447
tff(fact_10653_ATP_Olambda__448,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qd(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_448
tff(fact_10654_ATP_Olambda__449,axiom,
    ! [Uu2: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_qe(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_449
tff(fact_10655_ATP_Olambda__450,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yf(Uu2),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu2) ).

% ATP.lambda_450
tff(fact_10656_ATP_Olambda__451,axiom,
    ! [Uu2: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_cl(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_451
tff(fact_10657_ATP_Olambda__452,axiom,
    ! [Uu2: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_co(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_452
tff(fact_10658_ATP_Olambda__453,axiom,
    ! [Uu2: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cn(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_453
tff(fact_10659_ATP_Olambda__454,axiom,
    ! [Uu2: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_cq(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_454
tff(fact_10660_ATP_Olambda__455,axiom,
    ! [Uu2: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ck((Uu2)),Uua),(Uub))
    <=> ( ~ ( (Uub)
          <=> (Uu2) )
       => aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_455
tff(fact_10661_ATP_Olambda__456,axiom,
    ! [Uu2: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cp(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_456
tff(fact_10662_ATP_Olambda__457,axiom,
    ! [Uu2: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cm(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_457
tff(fact_10663_ATP_Olambda__458,axiom,
    ! [Uu2: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cj(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_458
tff(fact_10664_ATP_Olambda__459,axiom,
    ! [Uu2: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_lu(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_459
tff(fact_10665_ATP_Olambda__460,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mh(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_460
tff(fact_10666_ATP_Olambda__461,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ml(Uu2),Uua),Uub) = aa_real_real(plus_plus_real(Uu2),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_461
tff(fact_10667_ATP_Olambda__462,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_mi(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_462
tff(fact_10668_ATP_Olambda__463,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_463
tff(fact_10669_ATP_Olambda__464,axiom,
    ! [Uu2: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_mj(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_464
tff(fact_10670_ATP_Olambda__465,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_od(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_465
tff(fact_10671_ATP_Olambda__466,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_og(Uu2),Uua),Uub) = aa_nat_real(power_power_real(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_466
tff(fact_10672_ATP_Olambda__467,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oe(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_467
tff(fact_10673_ATP_Olambda__468,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_of(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_468
tff(fact_10674_ATP_Olambda__469,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xu(Uu2),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_469
tff(fact_10675_ATP_Olambda__470,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ja(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_470
tff(fact_10676_ATP_Olambda__471,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jd(Uu2),Uua),Uub) = aa_nat_real(power_power_real(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_471
tff(fact_10677_ATP_Olambda__472,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_jb(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_472
tff(fact_10678_ATP_Olambda__473,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jc(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_473
tff(fact_10679_ATP_Olambda__474,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mm(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_474
tff(fact_10680_ATP_Olambda__475,axiom,
    ! [Uu2: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_qp(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_475
tff(fact_10681_ATP_Olambda__476,axiom,
    ! [Uu2: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_qo(Uu2),Uua),Uub) = aa_nat_real(power_power_real(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_476
tff(fact_10682_ATP_Olambda__477,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qq(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_477
tff(fact_10683_ATP_Olambda__478,axiom,
    ! [Uu2: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_qr(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_478
tff(fact_10684_ATP_Olambda__479,axiom,
    ! [Uu2: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_qs(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_479
tff(fact_10685_ATP_Olambda__480,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: code_integer,Uub: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aTP_Lamm_jw(Uu2),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_com4217951221296830281nteger(Uu2,Uub)) ) ).

% ATP.lambda_480
tff(fact_10686_ATP_Olambda__481,axiom,
    ! [Uu2: fun_complex_nat,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_kc(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_complex_nat(Uu2,Uub)) ) ).

% ATP.lambda_481
tff(fact_10687_ATP_Olambda__482,axiom,
    ! [Uu2: fun_complex_int,Uua: int,Uub: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aTP_Lamm_kf(Uu2),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_complex_int(Uu2,Uub)) ) ).

% ATP.lambda_482
tff(fact_10688_ATP_Olambda__483,axiom,
    ! [Uu2: fun_li3231056112175589690nteger,Uua: code_integer,Uub: list_nat] :
      ( aa_list_nat_o(aa_Cod7157825952861150518_nat_o(aTP_Lamm_ku(Uu2),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_lis9097500579518594299nteger(Uu2,Uub)) ) ).

% ATP.lambda_483
tff(fact_10689_ATP_Olambda__484,axiom,
    ! [Uu2: fun_se7586849273513198688nteger,Uua: code_integer,Uub: set_nat] :
      ( aa_set_nat_o(aa_Cod793699401815963408_nat_o(aTP_Lamm_kr(Uu2),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_set871585252387937313nteger(Uu2,Uub)) ) ).

% ATP.lambda_484
tff(fact_10690_ATP_Olambda__485,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_kx(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_set_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_485
tff(fact_10691_ATP_Olambda__486,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_jz(Uu2),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu2,Uub)) ) ).

% ATP.lambda_486
tff(fact_10692_ATP_Olambda__487,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_kl(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_487
tff(fact_10693_ATP_Olambda__488,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_ki(Uu2),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu2,Uub)) ) ).

% ATP.lambda_488
tff(fact_10694_ATP_Olambda__489,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ko(Uu2),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_int_int(Uu2,Uub)) ) ).

% ATP.lambda_489
tff(fact_10695_ATP_Olambda__490,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sp(Uu2),Uua),Uub) = aa_real_real(plus_plus_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_490
tff(fact_10696_ATP_Olambda__491,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_so(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_491
tff(fact_10697_ATP_Olambda__492,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sq(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_492
tff(fact_10698_ATP_Olambda__493,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sr(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_493
tff(fact_10699_ATP_Olambda__494,axiom,
    ! [Uu2: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_zu(Uu2),Uua),Uub) = aa_real_real(powr_real2(Uu2),Uua) ).

% ATP.lambda_494
tff(fact_10700_ATP_Olambda__495,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nc(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_495
tff(fact_10701_ATP_Olambda__496,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_496
tff(fact_10702_ATP_Olambda__497,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qy(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_497
tff(fact_10703_ATP_Olambda__498,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_lj(Uu2),Uua),Uub) = aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_498
tff(fact_10704_ATP_Olambda__499,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ib(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_499
tff(fact_10705_ATP_Olambda__500,axiom,
    ! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yt(Uu2),Uua),Uub)
    <=> aa_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_500
tff(fact_10706_ATP_Olambda__501,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_it(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_501
tff(fact_10707_ATP_Olambda__502,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qt(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_502
tff(fact_10708_ATP_Olambda__503,axiom,
    ! [Uu2: fun_Pr4796943543331071514_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gj(Uu2),Uua),Uub)
    <=> aa_Pro6150637604445964947_nat_o(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ) ).

% ATP.lambda_503
tff(fact_10709_ATP_Olambda__504,axiom,
    ! [Uu2: fun_Pr2602117396471556196at_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gk(Uu2),Uua),Uub) = aa_Pro4243173770044397973at_nat(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_504
tff(fact_10710_ATP_Olambda__505,axiom,
    ! [Uu2: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_gl(Uu2),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_505
tff(fact_10711_ATP_Olambda__506,axiom,
    ! [Uu2: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_gm(Uu2),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_506
tff(fact_10712_ATP_Olambda__507,axiom,
    ! [Uu2: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gn(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_507
tff(fact_10713_ATP_Olambda__508,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(Uu2),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu2)) ).

% ATP.lambda_508
tff(fact_10714_ATP_Olambda__509,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_ir(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_509
tff(fact_10715_ATP_Olambda__510,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ip(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(power_power_real(Uua),Uub))) ).

% ATP.lambda_510
tff(fact_10716_ATP_Olambda__511,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xz(Uu2),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),Uua)) ).

% ATP.lambda_511
tff(fact_10717_ATP_Olambda__512,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: code_integer,Uub: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aTP_Lamm_jx(Uu2),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_com4217951221296830281nteger(Uu2,Uub)) ) ).

% ATP.lambda_512
tff(fact_10718_ATP_Olambda__513,axiom,
    ! [Uu2: fun_complex_nat,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_kd(Uu2),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_complex_nat(Uu2,Uub)) ) ).

% ATP.lambda_513
tff(fact_10719_ATP_Olambda__514,axiom,
    ! [Uu2: fun_complex_int,Uua: int,Uub: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aTP_Lamm_kg(Uu2),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_complex_int(Uu2,Uub)) ) ).

% ATP.lambda_514
tff(fact_10720_ATP_Olambda__515,axiom,
    ! [Uu2: fun_li3231056112175589690nteger,Uua: code_integer,Uub: list_nat] :
      ( aa_list_nat_o(aa_Cod7157825952861150518_nat_o(aTP_Lamm_kv(Uu2),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_lis9097500579518594299nteger(Uu2,Uub)) ) ).

% ATP.lambda_515
tff(fact_10721_ATP_Olambda__516,axiom,
    ! [Uu2: fun_se7586849273513198688nteger,Uua: code_integer,Uub: set_nat] :
      ( aa_set_nat_o(aa_Cod793699401815963408_nat_o(aTP_Lamm_ks(Uu2),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_set871585252387937313nteger(Uu2,Uub)) ) ).

% ATP.lambda_516
tff(fact_10722_ATP_Olambda__517,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_ky(Uu2),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_set_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_517
tff(fact_10723_ATP_Olambda__518,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_ka(Uu2),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu2,Uub)) ) ).

% ATP.lambda_518
tff(fact_10724_ATP_Olambda__519,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_km(Uu2),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_519
tff(fact_10725_ATP_Olambda__520,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_kj(Uu2),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu2,Uub)) ) ).

% ATP.lambda_520
tff(fact_10726_ATP_Olambda__521,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kp(Uu2),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_int_int(Uu2,Uub)) ) ).

% ATP.lambda_521
tff(fact_10727_ATP_Olambda__522,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uc(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_522
tff(fact_10728_ATP_Olambda__523,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_523
tff(fact_10729_ATP_Olambda__524,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ub(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_524
tff(fact_10730_ATP_Olambda__525,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ua(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_525
tff(fact_10731_ATP_Olambda__526,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ra(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_526
tff(fact_10732_ATP_Olambda__527,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rc(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_527
tff(fact_10733_ATP_Olambda__528,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_re(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_528
tff(fact_10734_ATP_Olambda__529,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rg(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_529
tff(fact_10735_ATP_Olambda__530,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ri(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_530
tff(fact_10736_ATP_Olambda__531,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rb(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_531
tff(fact_10737_ATP_Olambda__532,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rd(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_532
tff(fact_10738_ATP_Olambda__533,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rf(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_533
tff(fact_10739_ATP_Olambda__534,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rh(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_534
tff(fact_10740_ATP_Olambda__535,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rj(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_535
tff(fact_10741_ATP_Olambda__536,axiom,
    ! [Uu2: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_ni(Uu2),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu2,Uuc)) ).

% ATP.lambda_536
tff(fact_10742_ATP_Olambda__537,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_la(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_537
tff(fact_10743_ATP_Olambda__538,axiom,
    ! [Uu2: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_lc(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_538
tff(fact_10744_ATP_Olambda__539,axiom,
    ! [Uu2: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_le(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_539
tff(fact_10745_ATP_Olambda__540,axiom,
    ! [Uu2: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lg(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_540
tff(fact_10746_ATP_Olambda__541,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_kz(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_541
tff(fact_10747_ATP_Olambda__542,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lb(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_542
tff(fact_10748_ATP_Olambda__543,axiom,
    ! [Uu2: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_ld(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_nat(Uua,Uuc),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_543
tff(fact_10749_ATP_Olambda__544,axiom,
    ! [Uu2: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] :
      aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_lf(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_int(Uua,Uuc),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_544
tff(fact_10750_ATP_Olambda__545,axiom,
    ! [Uu2: $o,Uua: fun_o_real,Uub: fun_o_real,Uuc: $o] :
      aa_o_real(aa_fun688608109223240316o_real(aa_fun1133276423545935902o_real(aTP_Lamm_lh((Uu2)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu2),aa_o_real(Uua,(Uuc)),aa_o_real(Uub,(Uuc))) ).

% ATP.lambda_545
tff(fact_10751_ATP_Olambda__546,axiom,
    ! [Uu2: $o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_li((Uu2)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu2),aa_o_rat(Uua,(Uuc)),aa_o_rat(Uub,(Uuc))) ).

% ATP.lambda_546
tff(fact_10752_ATP_Olambda__547,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rt(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_547
tff(fact_10753_ATP_Olambda__548,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rv(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_548
tff(fact_10754_ATP_Olambda__549,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_rs(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_int_real(Uua,Uuc),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_549
tff(fact_10755_ATP_Olambda__550,axiom,
    ! [Uu2: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ru(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_550
tff(fact_10756_ATP_Olambda__551,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aa_fun4268748527592579270omplex(aTP_Lamm_jn(Uu2),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu2,Uuc),aa_complex_complex(Uua,Uuc),aa_complex_complex(Uub,Uuc)) ).

% ATP.lambda_551
tff(fact_10757_ATP_Olambda__552,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_jg(Uu2),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu2,Uuc),aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_552
tff(fact_10758_ATP_Olambda__553,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_jh(Uu2),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu2,Uuc),aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_553
tff(fact_10759_ATP_Olambda__554,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ji(Uu2),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu2,Uuc),aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_554
tff(fact_10760_ATP_Olambda__555,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_jj(Uu2),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu2,Uuc),aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_555
tff(fact_10761_ATP_Olambda__556,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_jo(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu2,Uuc),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_556
tff(fact_10762_ATP_Olambda__557,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_jk(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu2,Uuc),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_557
tff(fact_10763_ATP_Olambda__558,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_jm(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu2,Uuc),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_558
tff(fact_10764_ATP_Olambda__559,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_jl(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu2,Uuc),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_559
tff(fact_10765_ATP_Olambda__560,axiom,
    ! [Uu2: fun_int_o,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_jp(Uu2),Uua),Uub),Uuc) = $ite(aa_int_o(Uu2,Uuc),aa_int_int(Uua,Uuc),aa_int_int(Uub,Uuc)) ).

% ATP.lambda_560
tff(fact_10766_ATP_Olambda__561,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ms(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mr(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_561
tff(fact_10767_ATP_Olambda__562,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mq(Uu2),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mp(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_562
tff(fact_10768_ATP_Olambda__563,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mo(Uu2),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mn(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_563
tff(fact_10769_ATP_Olambda__564,axiom,
    ! [Uu2: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_ul(Uu2),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          $ite(
            Uuc = zero_zero_nat,
            aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),
            $ite(Uuc = Uu2,one_one_Code_integer,zero_z3403309356797280102nteger) )),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_564
tff(fact_10770_ATP_Olambda__565,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_uk(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex($ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu2,one_one_complex,zero_zero_complex) )),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_565
tff(fact_10771_ATP_Olambda__566,axiom,
    ! [Uu2: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_un(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real($ite(
            Uuc = zero_zero_nat,
            aa_real_real(uminus_uminus_real,Uub),
            $ite(Uuc = Uu2,one_one_real,zero_zero_real) )),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_566
tff(fact_10772_ATP_Olambda__567,axiom,
    ! [Uu2: nat,Uua: rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_um(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,
          $ite(
            Uuc = zero_zero_nat,
            aa_rat_rat(uminus_uminus_rat,Uub),
            $ite(Uuc = Uu2,one_one_rat,zero_zero_rat) )),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_567
tff(fact_10773_ATP_Olambda__568,axiom,
    ! [Uu2: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_uj(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(
            Uuc = zero_zero_nat,
            aa_int_int(uminus_uminus_int,Uub),
            $ite(Uuc = Uu2,one_one_int,zero_zero_int) )),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_568
tff(fact_10774_ATP_Olambda__569,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ts(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_tr(Uu2),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_569
tff(fact_10775_ATP_Olambda__570,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_tf(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_te(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_570
tff(fact_10776_ATP_Olambda__571,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_td(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_tc(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_571
tff(fact_10777_ATP_Olambda__572,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_tb(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ta(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_572
tff(fact_10778_ATP_Olambda__573,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_sz(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_sy(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_573
tff(fact_10779_ATP_Olambda__574,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xi(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_574
tff(fact_10780_ATP_Olambda__575,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xg(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_575
tff(fact_10781_ATP_Olambda__576,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xe(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_576
tff(fact_10782_ATP_Olambda__577,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xf(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_577
tff(fact_10783_ATP_Olambda__578,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nm(Uu2),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uub)) ).

% ATP.lambda_578
tff(fact_10784_ATP_Olambda__579,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ns(Uu2),Uua),Uub),Uuc) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu2),Uuc))),aa_nat_real(power_power_real(Uu2),Uub)) ).

% ATP.lambda_579
tff(fact_10785_ATP_Olambda__580,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_no(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uub)) ).

% ATP.lambda_580
tff(fact_10786_ATP_Olambda__581,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nq(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uu2),Uub)) ).

% ATP.lambda_581
tff(fact_10787_ATP_Olambda__582,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wd(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_582
tff(fact_10788_ATP_Olambda__583,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mn(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu2)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_583
tff(fact_10789_ATP_Olambda__584,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mr(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uua),Uu2)),Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_584
tff(fact_10790_ATP_Olambda__585,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mp(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),Uu2)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_585
tff(fact_10791_ATP_Olambda__586,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tp(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uu2),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_586
tff(fact_10792_ATP_Olambda__587,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tn(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu2),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_587
tff(fact_10793_ATP_Olambda__588,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_to(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu2),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_588
tff(fact_10794_ATP_Olambda__589,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tj(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_589
tff(fact_10795_ATP_Olambda__590,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tm(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(power_power_real(Uu2),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_590
tff(fact_10796_ATP_Olambda__591,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tk(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_591
tff(fact_10797_ATP_Olambda__592,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_st(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_592
tff(fact_10798_ATP_Olambda__593,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tl(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_593
tff(fact_10799_ATP_Olambda__594,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nu(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu2),Uuc)) ).

% ATP.lambda_594
tff(fact_10800_ATP_Olambda__595,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nx(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(power_power_real(Uu2),Uuc)) ).

% ATP.lambda_595
tff(fact_10801_ATP_Olambda__596,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nv(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu2),Uuc)) ).

% ATP.lambda_596
tff(fact_10802_ATP_Olambda__597,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nw(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu2),Uuc)) ).

% ATP.lambda_597
tff(fact_10803_ATP_Olambda__598,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_598
tff(fact_10804_ATP_Olambda__599,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nt(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uu2),Uuc)),aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_599
tff(fact_10805_ATP_Olambda__600,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_np(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_600
tff(fact_10806_ATP_Olambda__601,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nr(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_601
tff(fact_10807_ATP_Olambda__602,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ny(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_602
tff(fact_10808_ATP_Olambda__603,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ob(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uu2),Uuc)),aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_603
tff(fact_10809_ATP_Olambda__604,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nz(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_604
tff(fact_10810_ATP_Olambda__605,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_oa(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_605
tff(fact_10811_ATP_Olambda__606,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_sf(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_606
tff(fact_10812_ATP_Olambda__607,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wi(Uu2),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,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_607
tff(fact_10813_ATP_Olambda__608,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wg(Uu2),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,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_608
tff(fact_10814_ATP_Olambda__609,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wk(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_609
tff(fact_10815_ATP_Olambda__610,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wm(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_610
tff(fact_10816_ATP_Olambda__611,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_zq(Uu2),Uua),Uub),Uuc)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_611
tff(fact_10817_ATP_Olambda__612,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_fg(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_real_real(plus_plus_real(aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_612
tff(fact_10818_ATP_Olambda__613,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_fq(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_613
tff(fact_10819_ATP_Olambda__614,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ey(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_real(plus_plus_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_614
tff(fact_10820_ATP_Olambda__615,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fi(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_615
tff(fact_10821_ATP_Olambda__616,axiom,
    ! [Uu2: set_o,Uua: fun_o_real,Uub: fun_o_real,Uuc: $o] :
      ( aa_o_o(aa_fun1386971751305049950un_o_o(aa_fun3246320876514483410un_o_o(aTP_Lamm_fa(Uu2),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu2)
        & ( aa_real_real(plus_plus_real(aa_o_real(Uua,(Uuc))),aa_o_real(Uub,(Uuc))) != zero_zero_real ) ) ) ).

% ATP.lambda_616
tff(fact_10822_ATP_Olambda__617,axiom,
    ! [Uu2: set_o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_fk(Uu2),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Uua,(Uuc))),aa_o_rat(Uub,(Uuc))) != zero_zero_rat ) ) ) ).

% ATP.lambda_617
tff(fact_10823_ATP_Olambda__618,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_fe(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_real_real(plus_plus_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_618
tff(fact_10824_ATP_Olambda__619,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_fo(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_619
tff(fact_10825_ATP_Olambda__620,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_fc(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_real_real(plus_plus_real(aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_620
tff(fact_10826_ATP_Olambda__621,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_fm(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_621
tff(fact_10827_ATP_Olambda__622,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ix(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_622
tff(fact_10828_ATP_Olambda__623,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_iy(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_623
tff(fact_10829_ATP_Olambda__624,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_sy(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_624
tff(fact_10830_ATP_Olambda__625,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_te(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu2,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_625
tff(fact_10831_ATP_Olambda__626,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ta(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_626
tff(fact_10832_ATP_Olambda__627,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_tr(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_627
tff(fact_10833_ATP_Olambda__628,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_tc(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_628
tff(fact_10834_ATP_Olambda__629,axiom,
    ! [Uu2: $o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_ho((Uu2)),Uua),Uub),Uuc)
    <=> ( (Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_629
tff(fact_10835_ATP_Olambda__630,axiom,
    ! [Uu2: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_gh(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_630
tff(fact_10836_ATP_Olambda__631,axiom,
    ! [Uu2: fun_o_o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_fz(Uu2),Uua),Uub),Uuc)
    <=> aa_o_o(Uu2,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_631
tff(fact_10837_ATP_Olambda__632,axiom,
    ! [Uu2: fun_o_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun428955956450291160at_nat(aTP_Lamm_ga(Uu2),Uua),Uub),Uuc) = aa_o_nat(Uu2,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_632
tff(fact_10838_ATP_Olambda__633,axiom,
    ! [Uu2: fun_o_4034989691794805934at_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aa_fun8042081135417832459at_nat(aTP_Lamm_gi(Uu2),Uua),Uub),Uuc) = aa_o_f1925640989104809311at_nat(Uu2,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_633
tff(fact_10839_ATP_Olambda__634,axiom,
    ! [Uu2: fun_o_3115027942823555909_nat_o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun5069079742692966690_nat_o(aTP_Lamm_gd(Uu2),Uua),Uub),Uuc) = aa_o_f8445886544701262710_nat_o(Uu2,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_634
tff(fact_10840_ATP_Olambda__635,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun5171641919285310978_nat_o(aTP_Lamm_gb(Uu2),Uua),Uub),Uuc)
    <=> aa_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ) ).

% ATP.lambda_635
tff(fact_10841_ATP_Olambda__636,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_gc(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_636
tff(fact_10842_ATP_Olambda__637,axiom,
    ! [Uu2: fun_na3970208099681795815_nat_o,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun2100511394839469148_nat_o(aTP_Lamm_ge(Uu2),Uua),Uub),Uuc) = aa_nat7695548938202568488_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_637
tff(fact_10843_ATP_Olambda__638,axiom,
    ! [Uu2: fun_fu6891447142836604759at_o_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun7237989844890293688_nat_o(aTP_Lamm_gf(Uu2),Uua),Uub),Uuc)
    <=> aa_fun8408151826566693824at_o_o(Uu2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_638
tff(fact_10844_ATP_Olambda__639,axiom,
    ! [Uu2: fun_fu1115759055147092071_o_nat,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun3680403850923830236at_nat(aTP_Lamm_gg(Uu2),Uua),Uub),Uuc) = aa_fun1432809138467554728_o_nat(Uu2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_639
tff(fact_10845_ATP_Olambda__640,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_jf(Uu2),Uua),Uub),Uuc) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_640
tff(fact_10846_ATP_Olambda__641,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_je(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_641
tff(fact_10847_ATP_Olambda__642,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_qz(Uu2),Uua),Uub),Uuc) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_642
tff(fact_10848_ATP_Olambda__643,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yn(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_643
tff(fact_10849_ATP_Olambda__644,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yl(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_644
tff(fact_10850_ATP_Olambda__645,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yj(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_645
tff(fact_10851_ATP_Olambda__646,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yh(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_646
tff(fact_10852_ATP_Olambda__647,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_vg(Uu2),Uua),Uub),Uuc),Uud) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_vf(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_647
tff(fact_10853_ATP_Olambda__648,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_ve(Uu2),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_vd(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_648
tff(fact_10854_ATP_Olambda__649,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_vc(Uu2),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_vb(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_649
tff(fact_10855_ATP_Olambda__650,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_va(Uu2),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_uz(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_650
tff(fact_10856_ATP_Olambda__651,axiom,
    ! [Uu2: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_xh(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xg(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))))) ).

% ATP.lambda_651
tff(fact_10857_ATP_Olambda__652,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_uw(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_652
tff(fact_10858_ATP_Olambda__653,axiom,
    ! [Uu2: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_uy(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_653
tff(fact_10859_ATP_Olambda__654,axiom,
    ! [Uu2: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ux(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_654
tff(fact_10860_ATP_Olambda__655,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ud(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu2)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_655
tff(fact_10861_ATP_Olambda__656,axiom,
    ! [Uu2: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_uh(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uu2)),Uua)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_656
tff(fact_10862_ATP_Olambda__657,axiom,
    ! [Uu2: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uf(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uu2)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_657
tff(fact_10863_ATP_Olambda__658,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ue(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_658
tff(fact_10864_ATP_Olambda__659,axiom,
    ! [Uu2: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ui(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_659
tff(fact_10865_ATP_Olambda__660,axiom,
    ! [Uu2: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ug(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Uua)),Uud)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_660
tff(fact_10866_ATP_Olambda__661,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_uz(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_661
tff(fact_10867_ATP_Olambda__662,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_vf(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_662
tff(fact_10868_ATP_Olambda__663,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_vb(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_663
tff(fact_10869_ATP_Olambda__664,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_vd(Uu2),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_664
tff(fact_10870_ATP_Olambda__665,axiom,
    ! [Uu2: 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_yv(Uu2),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))),Uu2))
        & ! [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,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X4)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_665
tff(fact_10871_ATP_Olambda__666,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hp,Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_666
tff(fact_10872_ATP_Olambda__667,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_nd,Uu2) = Uu2 ).

% ATP.lambda_667
tff(fact_10873_ATP_Olambda__668,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_mg,Uu2) = Uu2 ).

% ATP.lambda_668
tff(fact_10874_ATP_Olambda__669,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_om,Uu2) = Uu2 ).

% ATP.lambda_669
tff(fact_10875_ATP_Olambda__670,axiom,
    ! [Uu2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_cf,Uu2)
    <=> $false ) ).

% ATP.lambda_670
tff(fact_10876_ATP_Olambda__671,axiom,
    ! [Uu2: complex] :
      ( aa_complex_o(aTP_Lamm_cc,Uu2)
    <=> $false ) ).

% ATP.lambda_671
tff(fact_10877_ATP_Olambda__672,axiom,
    ! [Uu2: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ce,Uu2)
    <=> $false ) ).

% ATP.lambda_672
tff(fact_10878_ATP_Olambda__673,axiom,
    ! [Uu2: $o] :
      ( aa_o_o(aTP_Lamm_cg,(Uu2))
    <=> $false ) ).

% ATP.lambda_673
tff(fact_10879_ATP_Olambda__674,axiom,
    ! [Uu2: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_cd,Uu2)
    <=> $false ) ).

% ATP.lambda_674
tff(fact_10880_ATP_Olambda__675,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_ch,Uu2)
    <=> $false ) ).

% ATP.lambda_675
tff(fact_10881_ATP_Olambda__676,axiom,
    ! [Uu2: int] :
      ( aa_int_o(aTP_Lamm_ci,Uu2)
    <=> $false ) ).

% ATP.lambda_676
tff(fact_10882_ATP_Olambda__677,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_vr,Uu2)
    <=> $true ) ).

% ATP.lambda_677

% Helper facts (18)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X3: $o,Y: $o] :
      ( ( (X3) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X3)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X3: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X3)),(Y))
      | ( (X3) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X3: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] :
      ( ( X3 != Y )
      | aa_complex_o(aa_com5808801942499186519plex_o(fequal_complex,X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] :
      ( ~ aa_complex_o(aa_com5808801942499186519plex_o(fequal_complex,X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( X3 != Y )
      | aa_set_nat_o(fequal_set_nat(X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(fequal_set_nat(X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( X3 != Y )
      | aa_list_nat_o(fequal_list_nat(X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X3 != Y )
      | aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X3),Y)
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( X3 != Y )
      | aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X3),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X3),Y)
      | ( X3 = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    sa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2) ).

%------------------------------------------------------------------------------